Jérémy Castéra
email: jeremy.castera@univ-amu.fr

Claire Coiffard Marre

Margaret Chan

Kit Yok

Kezang Sherab

Maria Antonietta A Impedovo

Tago Sarapuu

Alice Delserieys-Pedregosa

Sufiana Khatoon Malik

Hélène Cheneval-Armand

Claire Coiffard Marre

Alice Delserieys Pedregosa

Hélène Armand

Self-reported TPACK of teacher educators across six countries in Asia and Europe

Keywords: TPACK, cross-national study, university teachers, CFA

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Recent literature highlighted the importance of developing teachers' technological pedagogical and content knowledge (TPACK) for technology integration in teacher training programmes [START_REF] Byker | Examining Elementary Education Tearchers and Preservice Teachers' Self-Efficacy Related to Technological Pedagogical and Content Knowledge (TPACK)[END_REF][START_REF] Joo | Factors influencing preservice teachers' intention to use technology: TPACK, Teacher Self-efficacy, and Technology Acceptance Model[END_REF][START_REF] Chen | Exploring the Relationship Between Self-Regulation and TPACK of Taiwanese Secondary In-Service Teachers[END_REF]etc.). Indeed, in the 21st century, an effective teacher needs to know how to integrate technology into teaching [START_REF] Niess | Investigating TPACK: Knowledge growth in teaching with technology[END_REF][START_REF] Benton-Borghi | A Universally Designed for Learning (UDL) infused Technological Pedagogical Content Knowledge (TPACK) practitioners' model essential for teacher preparation in the 21st Century[END_REF]. The TPACK model proposed by [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] can be used to investigate teachers' knowledge for improving teacher training. The study is focused on teacher educators' TPACK as one of the most important elements in schoolteacher training. It has been shown that most educators believed that the successful integration of technology into classrooms is an easy task requiring merely technology skills [START_REF] Benson | Teaching with technology: Using TPACK to understand teaching expertise in online higher education[END_REF][START_REF] Schmidt | TPCK in K-6 literacy education: It's not that elementary[END_REF]. However, developing TPACK is fundamental for teachers and educators to implement technology in courses not merely focused on technology skills. [START_REF] Voogt | TPACK in teacher education: are we preparing teachers to use technology for early literacy?[END_REF], stated that 'it enables teachers to select and use hardware and software, identify the affordances (or lack thereof) of specific features and use the tools in pedagogically appropriate and effective ways. In this study, the seven-factor model based on [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] was used to analyse 574 teacher educators across six countries. A preliminary analysis indicated the results were consistent with this theoretical model. Thus, the objective was to investigate the possible links between educators' TPACK perception and demographic factors, such as gender, age, level of academic degree and country ("demographic factors" is used in its widely accepted definition, e.g. [START_REF] Baturay | An investigation of the impact of demographics on cyberloafing from an educational setting angle[END_REF][START_REF] Griffiths | Demographic factors and playing variables in online computer gaming[END_REF].

Theoretical background

2.1. TPACK model [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] theorised the TPACK model by adding a third dimension related to technological knowledge (TK) to Shulman's model [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF], which was based on two dimensions related to pedagogical knowledge (PK) and content knowledge (CK). Consequently, the three overlapping dimensions were formed logically into a seven-factor model, taking into account these three basic dimensions (TK, PK, and CK) and the interactions among them: technological content knowledge (TCK), technological pedagogical knowledge (TPK), pedagogical content knowledge (PCK), and technological pedagogical content knowledge (TPCK). This theoretical framework emerged from many empirical works and practical applications [START_REF] Koehler | Teachers learning technology by design[END_REF][START_REF] Koehler | With a little help from your students: A new model for faculty development and online course design[END_REF][START_REF] Koehler | Tracing the development of teacher knowledge in a design seminar: Integrating content, pedagogy and technology[END_REF]). However, other empirical studies failed to support these seven initial factors. [START_REF] Luik | Perceptions of technological, pedagogical and content knowledge (TPACK) among pre-service teachers in Estonia[END_REF] reported that the model measured three main factors within a preservice teacher sample: technology (T-factors, grouping all the factors linked to technology: TK, TPK, TCK, and TPACK), pedagogy (PK and PCK), and content (CK). Similarly, [START_REF] Cubeles | The effective integration of ICTs in universities: the role of knowledge and academic experience of professors[END_REF] validated a threefactor model with a sample of 113 university teachers: TK as the first factor, PK/PCK as the second, and TCK/TPK/TPCK as the third. They concluded that the number and the type of TPACK factors depended on the investigated population. This work is fully compliant with the study of [START_REF] Schmidt | Technological pedagogical content knowledge (TPACK) the development and validation of an assessment instrument for preservice teachers[END_REF], who found that Tfactors were merged in only one T-factor except TK, which was still an independent factor in their study. Furthermore, [START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF] described another type of construct: a five-dimension model consisting of TK, CK, knowledge of pedagogy, knowledge of teaching with technology, and knowledge from critical reflection for a Singaporean preservice-teacher sample. The main pitfall of the seven-factor model is the construct validity dependence regarding the tool used (generally a questionnaire) and the population sampled (which is context dependent). [START_REF] Archambault | Revisiting technological pedagogical content knowledge: Exploring the TPACK framework[END_REF] highlighted the challenge of creating an instrument which can be generalizable to every context and every subject-matter to be taught. Considering teaching as a complex and illstructured domain [START_REF] Koehler | What is technological pedagogical content knowledge (TPACK)? Contemporary Issues in Technology and Teacher Education[END_REF], the Mishra and Koehler's ideal model can be adapted to each learning context minimizing risk of misinterpretation of the results through of a non-adequate model. This precaution is usually taken by researchers in order to provide an adequate response to the learning situation investigated in terms of training remediation for instance. Therefore, clarifying the structure of the model is greatly recommended for any new population and context investigated, especially when a cross-national sample is investigated.

There are alternative or complementary models to TPACK that can be used to analyse technology integration in pedagogical contexts. The SAMR model describes technology integration in pedagogical context as a fourlevel-framework starting from the lowest efficiency (substitution) to highest (redefinition) of integration [START_REF] Puentedura | SAMR and TPCK : An Introduction[END_REF]. Another framework, probably even more interesting in terms of research potentialities, is the TPR & A (Technological Pedagogical Reasoning and Action: [START_REF] Harris | If There's TPACK, is There Technological Pedagogical Reasoning and Action[END_REF]. This framework is based on the Shulman's model [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] of Pedagogical Reasoning and Action (PR & A) which argues that any educational decision and action is related to knowledge-based PR & A. The knowledge dimensions of PR & R are comprehension, transformation (including preparation, representation, selection, and adaptation/tailoring to student characteristics), reflection, and new comprehension [START_REF] Harris | If There's TPACK, is There Technological Pedagogical Reasoning and Action[END_REF]. [START_REF] Hofer | Differentiating TPACK development: Using learning activity types with inservice and preservice teachers[END_REF] demonstrated the high potentiality of this idea by reaching the challenge of developing in-service and pre-service teachers' TPACK through that they called Learning Activity Types (LAT) which is based on the TPR & A model. However, the TPACK model was chosen as the framework guiding this particular study to ensure the best fitting with the purpose of the research goals. [START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF] revealed the importance of the gender factor in the TPACK perceptions of pre-service students. This study is consistent with other studies like that of [START_REF] Markauskaite | Gender issues in preservice teachers' training: ICT literacy and online learning[END_REF], who showed that male teachers are more confident in using computers than their female colleagues; [START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF], who found higher PK for female teachers but lower TK; and [START_REF] Scherer | On the quest for validity: Testing the factor structure and measurement invariance of the technology-dimensions in the Technological, Pedagogical, and Content Knowledge (TPACK) model[END_REF], who found that for all the T-factors, males report higher competency than females. However, thinking gender as predictor of TPACK can be interpreted as an essentialist view of the knowledge determinants. Obviously, there is no biological reason to explain the observed knowledge differences between males and females. The differences are rooted in social context where males and females are placed in. Generally, studies report gender differences without giving any explanation giving more room for essentialist interpretation. One goal of this study was to see if males and females are different in their self-reporting TPACK and if it is possible to relate the differences to other dependent demographic factors. Knowledge dimensions, including technology, are dependent on demographic factors, such as age and teaching experience. A negative correlation between age and dimensions including technology has been confirmed [START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF][START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF][START_REF] Lee | Exploring teachers' perceived self efficacy and technological pedagogical content knowledge with respect to educational use of the World Wide Web[END_REF]. Nevertheless, [START_REF] Guo | Digital natives, digital immigrants: An analysis of age and ICT competency in teacher education[END_REF] reported that there was no statistically significant difference with respect to technology competence among different age-groups of preservice teachers. Nevertheless, [START_REF] Hofer | Differentiating TPACK-based learning materials for preservice and inservice teachers[END_REF] highlighted the need of differentiating preservice TPACK-based training module for more experimented teachers. This is in complete agreement with the work of [START_REF] Lee | Exploring teachers' perceived self efficacy and technological pedagogical content knowledge with respect to educational use of the World Wide Web[END_REF] or [START_REF] Yaghi | Subject matter as a factor in educational computing by teachers in international settings[END_REF] who showed empirically that more teachers have experience, lower they are confident with their TPACK. Age and teaching experience are two strongly dependant factors which are difficult to clearly separate: the very large majority of experienced teachers are at the same time the oldest. Notwithstanding, a factor generally not tested in TPACK studies is how the level of academic degree of teachers/educators can influence TPACK perceptions. Considering this lack of information, another goal of this study was to make this possible relationship clearer. Moreover, the targeted samples usually included mainly pre-service and in-service teachers (e.g., [START_REF] Bueno-Alastuey | Can telecollaboration contribute to the TPACK development of pre-service teachers?[END_REF][START_REF] Cengiz | The development of TPACK, technology integrated self-efficacy and instructional technology outcome expectations of pre-service physical education teachers[END_REF][START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF][START_REF] Lee | Exploring teachers' perceived self efficacy and technological pedagogical content knowledge with respect to educational use of the World Wide Web[END_REF][START_REF] Luik | Perceptions of technological, pedagogical and content knowledge (TPACK) among pre-service teachers in Estonia[END_REF][START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF], and very few studies have investigated university teachers (e.g., [START_REF] Cubeles | The effective integration of ICTs in universities: the role of knowledge and academic experience of professors[END_REF]. Therefore, this study aims to clarify the structure of the TPACK model based on an international university teacher sample and to report the elements influencing TPACK, such as national context, gender, level of academic degree and age.

Demographic factors influencing individual TPACK

As suggested by [START_REF] Khine | Exploring relationships among TPACK constructs and ICT achievement among trainee teachers[END_REF], many studies have been carried out in different national contexts, but very few of them investigated the national and cultural context as a possible element influencing TPACK. For instance, [START_REF] Chang | Assessing university students' perceptions of their P hysics instructors' TPACK development in two contexts[END_REF] explored the TPACK perceptions of two physics instructors from China and Taiwan. They found TPACK differences attributed to the national teaching context, but this case study is not generalisable. [START_REF] Chai | Validating and modelling technological pedagogical content knowledge framework among Asian preservice teachers[END_REF] reported that the TPACK results of pre-service teachers from China, Hong Kong, Singapore, and Taiwan were dependent on the national context, but without specifying the findings. Overall, many researchers have investigated TPACK in separate national contexts and rarely from a cross-national perspective, meaning that comparatively little is known about the significance of the model in crossnational research contexts. This lack of data is probably rooted into the interpretation issues of cross-national studies. Given some relevant elements of countries' contexts is necessary in order to formulate explaining hypotheses and results' interpretation. Thus, the French tradition of "didactique", which is based on a framework very similar to the PCK model of Shulman, has been very present into the teacher training since decades reflecting (the work of [START_REF] Chevallard | La transposition didactique : du savoir savant au savoir enseigné[END_REF]. More generally, in Europe, research in education is quite old, for instance The European Association for Research on Learning and Instruction (EARLI) was born in 1985. Concerning the Asian countries, the context is much more different: in Bhutan (VanBalkom and Sherman 2010) and in Pakistan (Khatoon Malik and Tanzeela 2012) researchers noticed major problems such as the low quality of teacher training programs, lack of research and innovations. Nonetheless, Malaysia is very active considering educational innovations, for instance the Universiti Teknologi MARA has embarked on i-Learn Center since 2005 to encourage Blended learning in the educational system. In addition, since 2018, the programme 'Week without Walls @ UiTM Sarawak' provides option to lecturers to implement 100% online studies with experts from industries or other universities. Furthermore, all the institutions involved in this study are part of an international blended learning project since 2016 (BLTeae: Blended Learning for teacher Educators in Asia and Europe) indicating an increasing will of developing teacher educators' skills in Asia and Europe. Another element which can be very informative is the "use sub-index" of the IDI (ICT development index) that gives information about the population level about how they are able to use ICT. The ranking separate clearly European countries from Asian countries of this sample: Denmark (1 st ) Estonia (15 th ), France (17 th ), Malaysia (48 th ), Bhutan (114 th ) and Pakistan (152 nd )1 . This indicator is not specifically related to teachers, but it gives an opportunity to appreciate a general trend.

Rationale and research questions

The original TPACK seven-factor model of [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] has recently been called into question by empirical studies. It seems that the number of factors is linked to some specific contexts of studies. In this respect, this study investigated the robustness of the seven-factor model in the context of international comparison. The constancy of the seven-factor model in a cross-national perspective would give some support for using this model in further cross-national investigation. Moreover, despite the profusion of studies was based on primary and secondary school teachers, there is a lack of knowledge about factors influencing university teachers' TPACK. Hence, this study aims to explore this field by attempting to answer the following questions:

1. Are university teachers' perceptions of technological, pedagogical, and content knowledge in six countries compatible with the seven-factor TPACK model? 2. To what extent are the seven TPACK factors related to university teachers' demographic factors? The hypotheses were that demographic factors, often reported at local level (gender, age and level of academic degree), would affect TPACK perception of an international teacher educators' sample. However, it was assumed that the national context would contribute globally more than the other factors to their TPACK perception.

Methodology

Instrument

The survey tool is based on the work of [START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF], who developed a questionnaire for assessing pre-and in-service Singaporean science teachers' perceptions of technological pedagogical content knowledge. The questionnaire composed of different sections aiming to measure various constructs developed into the original seven-factor model [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF]): TK, PK, CK, TPK, TCK, PCK, and TPCK (Table 1). The questionnaire was originally developed to assess science teachers' TPACK perception. Thus, a few changes were made in order to extend the sample to non-science teacher educators. For instance, in the original tool, the first item was: I have sufficient knowledge of science. In the newly extended questionnaire, this item has been modified: I have sufficient knowledge of the subject matter I teach. A section of the questionnaire was devoted to collecting demographic factors: age (20-29, 30-39, 40-49, 50-59, 60-69, or 70+), gender, and highest academic level (high school, bachelor, master, or PhD). After the first modifications, the questionnaire was piloted in all the countries on a small sample of teacher educators to verify its comprehension. It was followed by short interviews with the piloted respondents to identify possible issues. However, no change was made to items during this phase, since the respondents estimated the item comprehension as satisfactory. The online questionnaire was administered in English by email in all the institutions involved, except in France, where it was translated and thereafter validated with back-translation before the final sampling. The translation was made in France in order to ensure a high response rate in an institution where English is rarely used to disseminate information.

Participants

The sample consisted of 574 university teachers providing courses for students trained to be primary or secondary school teachers from eight institutions across six countries. The institutions were the Royal University of Bhutan (Bhutan), Alborg University (Denmark), Tallinn University (Estonia), the Université d'Aix-Marseille (France), the Universiti Teknologi MARA (Malaysia), the Institut Pendidikan Guru (Malaysia), the National University of Modern Languages (Pakistan), and the International Islamic University (Pakistan). All the institutions had been selected because they were all involved in different initiatives using educational technology (such MOOC, Blended-learning, etc.). Moreover, all these institutions formed the consortium participating in the BLTeae project (Blended Learning for Teacher Educators in Asia and Europe) funded by the European Commission. As anticipated and due to the noticeable attention taken to select the participating institutions, more than 98% of the sample reported the use of technologies at least once a day. The institutions were very similar in terms of trained student profiles. The Royal University of Bhutan offers a four-year Bachelor of Education and one-year postgraduate programmes. Post-graduate programmes to train teachers are as well offered in France and in Estonia (Master level) whereas, in Pakistan, the institutions offer four-year bachelor programmes for the students like in Denmark and Malaysia. In addition, in Malaysia, it is possible to apply for an additional one-year postgraduate programme.

Sampling scheme and data collection

The questionnaire was administered following the same protocol in all institutions. The university teachers were invited through emails to respond to an online survey. All the data were collected anonymously between June 2017 and December 2017.

To address the research questions, the TPACK section, aiming to measure respondents' perceptions about TPACK, and the personal data section (e.g., age, gender, and academic background) were analysed and interpreted. The TPACK section had already been validated, since it was adapted from the work of [START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF] to be in ad equation with a large range of university teachers' profiles, with only one question excluded from the original questionnaire. The 26 items were based on a four-point Likert scale (from strongly disagree to strongly agree) and investigated seven factors of the TPACK model as independent scales, as initially defined by [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF]. Table 1 shows items and their factors' contributions. The raw Cronbach alphas were calculated for each factor and showed good internal consistency (> 0.8), except for the PCK factor (between 0.6 and 0.7), which was, however, acceptable [START_REF] Manerikar | Cronbach's alpha. A Peer review research journal[END_REF]. Specifically, these results support the seven-factor model with an acceptable reliability for each scale.

Statistical analysis

Based on the theoretical framework of TPACK [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF], the hypothesis of the perception of TPACK was structured according to the seven factors of the model. The goal was to test whether the collected data were consistent with the nature of the theoretical model. Accordingly, the fitting of the model was measured with confirmatory factor analysis (CFA) using the robust maximum likelihood method. The classic indicators were used to evaluate the model's goodness of fit: the relative chi-square, since the sample was greater than 200 (χ2/df < 5; [START_REF] Wheaton | Assessing reliability and stability in panel models[END_REF]; the root mean square error of approximation (RMSEA <0.08; [START_REF] Browne | Alternative ways of assessing model fit[END_REF]; the adjusted goodness of fit index (AGFI > 0.9; [START_REF] Tanaka | A fit index for covariance structure models under arbitrary GLS estimation[END_REF]; the standardised root mean square residual (SRMR <0.05; Cangur and Ercan 2015); and the comparative fit index (CFI >0.9; Bentler 1990). To address the objectives of the second research question, Kruskal-Wallis chi-squared tests were performed, since the measurements failed to meet the normality assumption. This was completed by post-hoc Wilcoxon signed rank tests to compare with adjusted p-values [START_REF] Holm | A simple sequentially rejective multiple test procedure[END_REF], taking into account the problem of multiple comparisons. In addition, an ANOVA was performed to investigate the relative impact of the national context comparatively to the other factors tested (level of academic degree, gender, age and interactions between them). All the statistical analyses were computed through the use of the R software.

Results

CFA confirming the seven-factor model

CFA using the robust maximum likelihood method was performed on the TPACK variables in order to measure the seven-factor model's fitness. The CFA indicators are reported in Table 2. The chi-square value was significant; however, the sensitivity of the chi-square test to the large sample size (> 200) made this result no longer relevant [START_REF] Schermelleh-Engel | Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures[END_REF], and the relative chi-square value was acceptable (χ2/df=4.2; less than 5). The other indicators indicated good fit: RMSEA <0.08 [START_REF] Browne | Alternative ways of assessing model fit[END_REF]; AGFI >0.9 [START_REF] Tanaka | A fit index for covariance structure models under arbitrary GLS estimation[END_REF]; SRMR <0.05 [START_REF] Cangur | Comparison of model fit indices used in structural equation modeling under multivariate normality[END_REF]; CFI >0.9 [START_REF] Bentler | Comparative fit indexes in structural models[END_REF].

TPACK perceptions

Country differences

The findings supported a relatively similar trend among countries. In each country the lower scores were generally related to T-factors (Table 3).

Table 3: Seven factors' means among countries (all the factors have been rescaled to be out of 100). The significance is based on Kruskal-Wallis chi-squared tests, completed by post-hoc Wilcoxon signed rank tests.

However, there were strong differences between countries (Table 3) related to five factors: PCK (χ 2 (5) = 28.74); TK (χ 2 (5) = 34.40); TPK (χ 2 (5) = 48.97); TCK (χ 2 (5) = 46.16); TPC (χ 2 (5) = 33.38). The differences were mainly related to the scores of the Bhutanese sample, which were lower than many of the other countries. More specifically and concerning the PCK factor, the post-hoc Wilcoxon signed rank test indicated that the sample from Bhutan had significantly lower results regarding PCK than Denmark (p =0.026), France (p = 0.000), and Malaysia (p = 0.022), whereas the French sample had significantly higher results than Estonia (p = 0.047) and Pakistan (p = 0.003). The same trend was observed considering the TK, TPK, TCK, and TPC factors: Bhutan had lower scores than the majority of countries. As shown in Table 3, the Bhutanese sample showed significantly lower results regarding TK factors compared to Estonia (p = 0.000), Malaysia (p = 0.007), and Pakistan (p = 0.000). It was also observed that on the TPK scale, the French sample (M = 67.69) had the second lowest score after Bhutan, which was significantly lower than Pakistan's score (p = 0.038).

Age differences

Age, among the demographic factors, was hypothesised to be related to TPACK perceptions. Table 4 illustrates a positive and gradual evolution of mean scores of factors CK, PK, and PCK from the youngest age group to the oldest age group. There were score differences of 8% (CK), 5% (PK), and 7% (PCK) between the youngest group and the oldest group. The Kruskal-Wallis chi-squared tests showed significant variations between age groups for CK (p = 0.025) and PCK (p = 0.018). Nevertheless, the post-hoc test did not show any significant difference between groups. Conversely, it was indicated that factors including technology (TK, TPK, TCK, and TPC) gradually decreased from the youngest group to the oldest (significant Kruskal-Wallis tests). The difference between groups was especially important for the TK scores, in which the two youngest groups' scores were significantly higher (post-hoc tests). Other post-hoc multiple comparisons showed a significantly lower score for the oldest group (compared to 1, 2, and 4) and lower TCK score of Group 4 compared to Group 2. There was no difference between age groups concerning the TPC scores.

Other demographic factors and the global influence of countries' contexts

Two other demographic variables were tested according to the seven factors of the TPACK model: level of academic degree and gender. None of these comparisons revealed significant differences between tested groups. Moreover, a normalized TPACK score was computed and an ANOVA was performed on the four factors investigated in this study: age, level of academic degree, gender and country. The use of ANOVA was permitted since the conditions of this test were verified (normality, homoscedasticity and independence of residuals). For the normality a Shapiro test obtained W=0.99 with the p-value=0.78, for the homoscedasticity the Breusch-Pagan test obtained BP=19.05 and p-value=0.09 and for the independence Durbin Watson Test obtained DW=1.84 with p-value=0.052. The ANOVA showed that the country context has a significant effect on the TPACK score (p-value<0.001). None of the other factors or interactions between factors revealed significant differences.

Discussion

Findings

1. The first major result regards the substantial validation of the seven-factor TPACK model across a sixcountry sample (research question 1). The seven-factor model elaborated by [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] was supported by the results of the CFA. Other works were similar [START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF][START_REF] Schmidt | Technological pedagogical content knowledge (TPACK) the development and validation of an assessment instrument for preservice teachers[END_REF], showing consistency with the original seven-factor model in measuring perceptions of TPACK. The relative stability of the model structure across countries supports the relevance of using the seven-factor model when analysing cross-national surveys.

2. The second major result is the relative differences of university teachers' TPACK perceptions across six countries of Europe and Asia (research question 2). The score differences for the seven TPACK factors were mainly related to the Bhutanese sample. If it was excluded, there were only a few differences in PCK, where France had a significantly higher score compared to Estonia and Pakistan, and in TPK, where Pakistan had a significantly higher score than France. There were no significant differences for the other factors. The score of the French university teachers regarding PCK can be attributed to the long tradition of the French didactique in teacher training. However, the low TPK score of the French sample which was significant compared to Pakistan, revealed a lagging technology integration in the French educational system, which has been admitted by the government [START_REF] Karayan | Le retard français dans l'école numérique a été utile[END_REF]). In the case of Bhutan, teacher education is faced with many issues, such as teachers' low professional esteem (VanBalkom and Sherman 2010), and there has already been a call for revamping the teacher preparation programmes to provide more pedagogical and content-related knowledge as well as more ICT facilities (Ministry of Education 2014; [START_REF] Sherab | Efficacy of the Four-Year B. Ed Primary Programme at Paro College of Education[END_REF]. Many years later, this trend is still relevant through TPACK perception which showed significantly lowest scores in PCK, TK, TPK, TCK, and TPC. However, in every country it is possible to notice a similar trend: T-factors were usually less rated. This result clearly highlights the training need of teacher educators not only on using technology (TK) but also on how to use it to support content and/or pedagogical knowledge (TCK, TPK, and TPC). The ranking based on the "use sub-index" of the IDI (ICT development index) fits well for the Denmark and Estonia which had generally better scores on dimensions including technologies. However, for the other countries, it is not clear. In the case of Pakistan and Malaysian, teacher educators had comparable or even sometimes the best scores on dimension including technologies. This difference with the IDI is probably rooted into the important gap in Pakistan and Malaysia between the level of teacher educators and the public and probably a less important gap in Europe.

3. The third major result is the dependence of age on TPACK factors (research question 2). The global comparisons of age-groups were almost all significant (except for PK scores), showing a clear relationship between age and perceptions of TPACK. However, the multiple pairwise comparisons failed to attain the statistical significance level. This a priori contradiction can be explained by the limit of the statistic and methodological approach. Indeed, the small sizes of Group 1 (the youngest) and Group 5 (the oldest) introduced a lack of statistical power less likely to detect significant differences. Another issue was the use of the Holm p-value adjustment to counteract the problem of multiple comparisons. Without any adjustment, it appeared that the level of significance was reached in many cases; for instance, the group of 30-39-year-old teachers had a higher significant TPC score compared to the three older groups. This adjustment came at the cost of increasing the type II error risk (not detecting the difference). However, much evidence converged in the sense that age and TPACK perceptions were strongly related, as suggested by [START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF], Lee andTsai (2010), and[START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF]. The recommendations that it can be formulated for the institutions involved into the research is to differentiate some teacher educators' training modules based on the specific lacks identified in each national context. Moreover, since age is another important factor influencing technology knowledge and how to integrate it into instruction, Learning Activity Types approach [START_REF] Hofer | Differentiating TPACK development: Using learning activity types with inservice and preservice teachers[END_REF] for building TPACK and differentiating experimented and unexperimented teacher educators would be an interesting framework for developing teacher educators' TPACK. Nevertheless, the ANOVA based on the TPACK global scores (not on the different seven sub-scales), showed that the "country context" is the only significant factor. This result supports that compared to other factors (especially age which is significant for some specific TPACK dimensions) the country context is the most important factors explaining differences on the TPACK . 4. The fourth major result is the independence of gender and level of training from the TPACK factors (research question 2). The results of the study also supported that there was no gender difference in the TPACK perception, in contrast with other works in which women demonstrated more difficulties with knowledge related to technology [START_REF] Koh | Examining the technological pedagogical content knowledge of Singapore pre-service teachers with a large-scale survey[END_REF][START_REF] Lin | Identifying Science Teachers' Perceptions of Technological Pedagogical and Content Knowledge (TPACK)[END_REF][START_REF] Markauskaite | Gender issues in preservice teachers' training: ICT literacy and online learning[END_REF]. In gender comparisons, context matters [START_REF] Yoder | Making gender comparisons more meaningful: A call for more attention to social context[END_REF]: in previous studies were conducted in Singapore and in Australia, whereas this study sample was from Europe, Pakistan, Malaysia, and Bhutan. In the last few years policymakers and researchers have made efforts to bridge the gender gap (e.g.; [START_REF] Krämer | Closing the gender gap in STEM with friendly male instructors? On the effects of rapport behavior and gender of a virtual agent in an instructional interaction[END_REF][START_REF] Reilly | Gender differences in spatial ability: Implications for STEM education and approaches to reducing the gender gap for parents and educators[END_REF][START_REF] Wang | Gender gap in science, technology, engineering, and mathematics (STEM): Current knowledge, implications for practice, policy, and future directions[END_REF]. The question therefore relates to whether the first effects of these new teaching strategies can be observed in this study? The absence of differences between the educators' initial level of training can be explained by the presence, in all the institutions, of skills development programmes contributing to the development of teaching capacities of the university staff. Thus, the lifelong training and skills development programmes allowed by universities contribute to reduce the professional skill differences among teacher educators.

Study limitations

The data collection was performed while trying to respect a sampling protocol. However, in some groups the number of participants was lower than expected. Since the sample size was low, it was difficult to detect significant differences in some instances; for example, detecting differences between the Estonian and Danish groups could be a limiting factor. Nevertheless, some differences were detected between the other countries, illustrating the context dependence of the TPACK levels of teacher educators. The current comparison study was limited using two versions of the questionnaire, one in French and one in English. Hence, a risk of concept mistranslation had been introduced decreasing the validly of comparisons. However, reasonable translation precautions (translation and backtranslation process) have been taken in order to limit this bias. Since all the Asian countries of this study adopt English as an official language, the choice of using the English version even in Estonia and in Denmark where English is not an official national language was made. However, the good level in English of those sampled populations, the habit of having English information disseminated by their institutions and the "easy-answer" questionnaire provided confidence in the validity of the results obtained.

Table 1 :

 1 Item contributions to the seven-factor TPACK model and Cronbach alphas

	Item	TPACK factor	Cronbach alpha
		scale

Table 2 :

 2 Goodness of fit indices

	χ2	df	p value	RMSEA	AGFI	SRMR	CFI
	1163.981	278.000	.000	0.075	0.965	0.048	0.917

Table 4 :

 4

		N	Mean
			CK	PK	PCK	TK	TPK	TCK	TPC
	Bhutan (Bh)	80 84.88 808	69.38	64.69	61.25	59.31	64.94

Seven factors' means among age groups (all the factors have been rescaled to be out of 100). The significance is based on Kruskal-Wallis chi-squared tests, completed by post-hoc Wilcoxon signed rank tests.

https://www.itu.int/net4/ITU-D/idi/2017/index.html#idi2017rank-tab

The project 'Blended Learning Courses for teacher educators between Asia and Europe' (n°: 574130-EPP-1-2016-1-FR-EPPKA2-CBHE-JP) is supported by the European Commission Education, Audiovisual and Culture Executive Agency (EACEA) -Erasmus+ Higher Education -International Capacity Building programme.


Feng Qi

Ai-Qi Liu

COMPLETELY MONOTONIC DEGREES FOR A DIFFERENCE BETWEEN THE LOGARITHMIC AND PSI FUNCTIONS

Keywords: 2010 Mathematics Subject Classification. Primary 26A48, Secondary 33B15, 44A10 complete monotonicity, completely monotonic degree, difference, psi function, logarithmic function, concise proof, gamma function, asymptotic formula, remainder, conjecture

In the paper, the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions, secondly compute completely monotonic degree of the above-mentioned function, and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical Euler gamma function.

Preliminaries

Recall from [16, Chapter XIII], [START_REF] Schilling | Bernstein Functions-Theory and Applications[END_REF]Chapter 1], and [32, Chapter IV] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and

(-1) n f (n) (x) ≥ 0, x ∈ I, n ∈ {0} ∪ N.
Theorem 12b in [32, p. 161] states that a necessary and sufficient condition for f (x) to be completely monotonic on (0, ∞) is that [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] f (x) = ∞ 0 e -xt d µ(t), x ∈ (0, ∞),

where µ(t) is non-decreasing and the above integral converges for x ∈ (0, ∞). In other words or simply speaking, a function is completely monotonic on (0, ∞) if and only if it is a Laplace transform of a non-negative measure.

In [2, pp. 374-375, Theorem 1] and [8, Theorem 1], it was verified that the function [START_REF] Alzer | On some inequalities for the gamma and psi functions[END_REF] x α [ln x -ψ(x)], α ∈ R is completely monotonic on (0, ∞) if and only if α ≤ 1, where ψ(z) = Γ (z) Γ(z) is called the psi function and Γ(z) denotes the classical Euler gamma function which can be defined [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF][START_REF] Guo | Two new proofs of the complete monotonicity of a function involving the psi function[END_REF][START_REF]NIST Handbook of Mathematical Functions[END_REF][START_REF] Qiu | Some properties of the gamma and psi functions, with applications[END_REF] by

Γ(z) = ∞ 0 t z-1 e -t d t, (z) > 0 or by Γ(z) = lim n→∞ n!n z n k=0 (z + k) , z ∈ C \ {0, -1, -2, . . . }.
Let f (x) be a completely monotonic function on (0, ∞) and denote f (∞) = lim x→∞ f (x) ≥ 0. When the function x α [f (x) -f (∞)] is completely monotonic on (0, ∞) if and only if 0 ≤ α ≤ r ∈ R, we say that the number r is the completely monotonic degree of f (x) with respect to x ∈ (0, ∞); if the function x α [f (x)-f (∞)] is completely monotonic on (0, ∞) for all α ∈ R, then we say that the completely monotonic degree of f (x) with respect to x ∈ (0, ∞) is ∞. For convenience, a notation deg x cm [f (x)] was designed to denote the completely monotonic degree r of f (x) with respect to x ∈ (0, ∞). For more information on completely monotonic degree and its properties, please refer to the papers [START_REF] Guo | A completely monotonic function involving the tri-gamma function and with degree one[END_REF][START_REF] Koumandos | Monotonicity of some functions involving the gamma and psi functions[END_REF][START_REF] Koumandos | Some completely monotonic functions of positive order[END_REF][START_REF] Koumandos | Complete monotonicity and related properties of some special functions[END_REF][START_REF] Koumandos | Absolutely monotonic functions related to Euler's gamma function and Barnes' double and triple gamma function[END_REF][START_REF] Koumandos | Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function[END_REF][START_REF] Qi | Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions[END_REF][START_REF] Qi | On complete monotonicity for several classes of functions related to ratios of gamma functions[END_REF][START_REF] Qi | Lévy-Khintchine representation of Toader-Qi mean[END_REF][START_REF] Qi | The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function[END_REF][START_REF] Qi | Integral representations and properties of some functions involving the logarithmic function[END_REF][START_REF] Qi | Integral representations of bivariate complex geometric mean and their applications[END_REF][START_REF] Qi | Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean[END_REF][START_REF] Qi | The harmonic and geometric means are Bernstein functions[END_REF] and closely related references therein.

The necessary and sufficient condition α ≤ 1 for the function ( 2) to be completely monotonic on (0, ∞) means that the completely monotonic degree of the completely monotonic function ln x -ψ(x) on (0, ∞) is

deg x cm [ln x -ψ(x)] = 1.
In [START_REF] Qiu | Some properties of the gamma and psi functions, with applications[END_REF]Theorem 1.7], it was proved that the function

x 2 [ψ(x) -ln x] + x 2
is strictly decreasing and convex on (0, ∞) and, as x → ∞, tends to -1 12 . In [3, Theorem 1], it was discovered that the function

(3) Φ(x) = x 2 [ψ(x) -ln x] + x 2 + 1
12 is completely monotonic on (0, ∞).

In the second section of this paper, motivated by the second proof of [8, Theorem 1] and with the aid of some conclusions in [START_REF] Zhang | A concise proof for properties of three functions involving the exponential function[END_REF], we will present a concise proof, which is simpler and shorter than the corresponding proof in [3, Theorem 1], for complete monotonicity of the function Φ(x) on (0, ∞).

In the third section of this paper, we will compute the completely monotonic degree of the completely monotonic function

(4) φ(x) = Φ(x) x 2 = ψ(x) -ln x + 1 2x + 1 12x 2 on (0, ∞).
In the fourth section of this paper, we will pose conjectures on completely monotonic degrees of remainders for the asymptotic formula of ln Γ(x) and for polygamma functions ψ (k) (x) for k ≥ 0.

Complete monotonicity of the function Φ(x)

The first main result in this paper can be stated as the following theorem.

Theorem 1. The function Φ(x) defined in (3) is completely monotonic on (0, ∞), with the limits

(5) lim x→0 + Φ(x) = 1 12 and lim x→∞ Φ(x) = 0.
Proof. In [18, p. 140, 5.9.13], it was listed that

ψ(z) = ln z + ∞ 0 1 t - 1 1 -e -t e -tz d t.
By integration by parts, this formula can be reformulated as

ψ(x) -ln x + 1 x = ∞ 0 1 t - 1 e t -1 e -xt d t = - 1 x h(t)e -xt t=∞ t=0 - ∞ 0 h (t)e -xt d t = 1 2x + 1 x ∞ 0 h (t)e -xt d t,
where the function

h(t) =      1 t - 1 e t -1 , t = 0 1 2 , t = 0 is convex on (0, ∞), with the limits (6) lim t→0 + h (t) = - 1 12 and lim t→∞ h (t) = 0.
For detailed information on the function h(t), please refer to [START_REF] Guo | A simple proof of logarithmic convexity of extended mean values[END_REF][START_REF] Guo | Properties and applications of a function involving exponential functions[END_REF][START_REF] Guo | The function (b x -a x )/x: Logarithmic convexity and applications to extended mean values[END_REF][START_REF] Qi | Two closed forms for the Bernoulli polynomials[END_REF] and closely related references therein. Consequently, integrating by parts again yields

Φ(x) = 1 12 + x ∞ 0 h (t)e -xt d t = 1 12 - ∞ 0 h (t) d e -xt = ∞ 0 h (t)e -xt d t.
Accordingly, by virtue of [32, p. 161, Theorem 12b] mentioned at the beginning of this paper and with the help of the convexity for the function h(t) on (0, ∞), we immediately see that the function Φ(x) is completely monotonic on (0, ∞), while, in light of the limits in (6), we readily derive the limits in [START_REF] Guo | A simple proof of logarithmic convexity of extended mean values[END_REF]. The proof of Theorem 1 is complete.

Completely monotonic degree of the function φ(x)

The second main result in this paper can be stated as the following theorem.

Theorem 2. The completely monotonic degree of the completely monotonic function φ(x) defined in (4) on (0, ∞) is

(7) deg x cm [φ(x)] = 2.
Proof. Since the relation Φ(x) = x 2 φ(x) and the function φ(x) is completely monotonic on (0, ∞), see [22, Theorem 1], then Theorem 1 in [START_REF] Chen | Some properties of functions related to the gamma and psi functions[END_REF] and Theorem 1 in this paper mean [START_REF] Guo | Two new proofs of the complete monotonicity of a function involving the psi function[END_REF] deg x cm [φ(x)] ≥ 2. On the other hand side, if x α φ(x) is completely monotonic on (0, ∞), then its first derivative is not positive, that is,

αx α-1 φ(x) + x α φ (x) ≤ 0 which can be rearranged as α ≤ - xφ (x) φ(x) = - x 3 ψ (x + 1) -1 x + 1 2x 2 -1 6x 3 x 2 [ψ(x) -ln x] + x 2 + 1 12 → 2
for x → 0 + , where we used the recursion relation ψ (x + 1) = ψ (x) -1 x 2 and the first limit in [START_REF] Guo | A simple proof of logarithmic convexity of extended mean values[END_REF]. This means that [START_REF] Guo | Sharp inequalities for polygamma functions[END_REF] deg x cm [φ(x)] ≤ 2. Combining ( 8) with [START_REF] Guo | Sharp inequalities for polygamma functions[END_REF] concludes that the completely monotonic degree of the function φ(x) on (0, ∞) is 2. The proof of Theorem 2 is complete.

Conjectures

The third main result in this paper is to pose conjectures on completely monotonic degrees of remainders for the asymptotic formula of ln Γ(x) and on completely monotonic degrees of derivatives of these remainders.

In [22, Theorem 1], among other things, the function φ(x) and three functions

ln x - 1 2x -ψ(x), ψ (x) - 1 x - 1 2x 2 - 1 6x 3 + 1 30x 5 , 1 x + 1 2x 2 + 1 6x 3 -ψ (x)
were proved to be completely monotonic on (0, ∞). What are the completely monotonic degrees of these three functions on (0, ∞)? We guess that deg x cm ln x -

1 2x -ψ(x) = 1, (10)
deg x cm 1 x + 1 2x 2 + 1 6x 3 -ψ (x) = 2, (11) deg x cm ψ (x) - 1 x - 1 2x 2 - 1 6x 3 + 1 30x 5 = 4. (12)
In [2, Theorem 8] and [START_REF] Koumandos | Remarks on some completely monotonic functions[END_REF]Theorem 2], the functions

(13) R n (x) = (-1) n ln Γ(x)-x- 1 2 ln x+x- 1 2 ln(2π)- n k=1 B 2k (2k -1)2k 1 x 2k-1
for n ≥ 0 were proved to be completely monotonic on (0, ∞). Consequently, the functions (-1) m [R n (x)] (m) for m, n ≥ 0 are completely monotonic on (0, ∞). See also [START_REF] Guo | Sharp inequalities for polygamma functions[END_REF]Section 1.4] and [START_REF] Mortici | Very accurate estimates of the polygamma functions[END_REF]Theorem 3.1]. The functions (-1) n R n (x) for n ≥ 0 are remainders for the asymptotic formula of ln Γ(x). See [1, p. 257, 6.1.40] and [18, p. 140, 5.11.1].

In [START_REF] Koumandos | Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function[END_REF]Theorem 2.1], the completely monotonic degree of the function R n (x) for n ≥ 0 on (0, ∞) was proved to be at least n. The equations ( 7), ( 10), [START_REF] Koumandos | Remarks on some completely monotonic functions[END_REF], and [START_REF] Koumandos | Some completely monotonic functions of positive order[END_REF] and plotting by the well-known software Wolfram Mathematica imply that

deg x cm (-1) 0 [R 0 (x)] (0) = 0, deg x cm (-1) 0 [R 1 (x)] (0) = 1, deg x cm (-1) 0 [R 2 (x)] (0) = 2, deg x cm (-1) 0 [R 3 (x)] (0) = 4, deg x cm (-1) 0 [R 4 (x)] (0) = 6, deg x cm (-1) 0 [R 5 (x)] (0) = 8, deg x cm (-1) 1 [R 0 (x)] (1) = 1, deg x cm (-1) 1 [R 1 (x)] (1) = 2, deg x cm (-1) 1 [R 2 (x)] (1) = 3, deg x cm (-1) 1 [R 3 (x)] (1) = 5, deg x cm (-1) 1 [R 4 (x)] (1) = 7, deg x cm (-1) 1 [R 5 (x)] (1) = 9, deg x cm (-1) 2 [R 0 (x)] (2) = 1, deg x cm (-1) 2 [R 1 (x)] (2) = 2, deg x cm (-1) 2 [R 2 (x)] (2) = 4, deg x cm (-1) 2 [R 3 (x)] (2) = 6, deg x cm (-1) 2 [R 4 (x)] (2) = 8, deg x cm (-1) 2 [R 5 (x)] (2) = 10, deg x cm (-1) 3 [R 0 (x)] (3) = 2, deg x cm (-1) 3 [R 1 (x)] (3) = 3, deg x cm (-1) 3 [R 2 (x)] (3) = 5, deg x cm (-1) 3 [R 3 (x)] (3) = 7, deg x cm (-1) 3 [R 4 (x)] (3) = 9, deg x cm (-1) 3 [R 5 (x)] (3) = 11, deg x cm (-1) 4 [R 0 (x)] (4) = 3, deg x cm (-1) 4 [R 1 (x)] (4) = 4, deg x cm (-1) 4 [R 2 (x)] (4) = 6, deg x cm (-1) 4 [R 3 (x)] (4) = 8, deg x cm (-1) 4 [R 4 (x)] (4) = 10, deg x cm (-1) 4 [R 5 (x)] (4) = 12, deg x cm (-1) 5 [R 0 (x)] (5) = 4, deg x cm (-1) 5 [R 1 (x)] (5) = 5, deg x cm (-1) 5 [R 2 (x)] (5) = 7, deg x cm (-1) 5 [R 3 (x)] (5) = 9, deg x cm (-1) 5 [R 4 (x)] (5) = 11, deg x cm (-1) 5 [R 5 (x)] (5) = 13
. What are the completely monotonic degrees of the completely monotonic functions (-1) m [R n (x)] (m) for m, n ≥ 0 on (0, ∞)? We conjecture that (i) when m = 0, the completely monotonic degrees of R n (x) on (0, ∞) satisfy

deg x cm [R 0 (x)] = 0, deg x cm [R 1 (x)] = 1, and deg x cm [R n (x)] = 2(n -1), n ≥ 2; (ii) when m = 1, the completely monotonic degrees of -R n (x) on (0, ∞) satisfy deg x cm [-R 0 (x)] = 1, deg x cm [-R 1 (
x)] = 2, and deg x cm [-R n (x)] = 2n -1, n ≥ 2; (iii) when m ≥ 2, the completely monotonic degrees of (-1) m [R n (x)] (m) satisfy

deg x cm (-1) m [R 0 (x)] (m) = m -1, deg x cm (-1) m [R 1 (x)] (m) = m, and
deg x cm (-1) m [R n (x)] (m) = m + 2(n -1), n ≥ 2.

Remarks

Finally we give several remarks on the significance and importance of results obtained in this paper.

Remark 1. The significance of completely monotonic functions can be seen from the above equation ( 1) cited from [START_REF] Widder | The Laplace Transform[END_REF]Theorem 12b].

Completely monotonic degree is a new notion introduced in very recent years. See [START_REF] Guo | A completely monotonic function involving the tri-gamma function and with degree one[END_REF][START_REF] Koumandos | Monotonicity of some functions involving the gamma and psi functions[END_REF][START_REF] Koumandos | Some completely monotonic functions of positive order[END_REF][START_REF] Koumandos | Complete monotonicity and related properties of some special functions[END_REF][START_REF] Koumandos | Absolutely monotonic functions related to Euler's gamma function and Barnes' double and triple gamma function[END_REF][START_REF] Koumandos | Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function[END_REF][START_REF] Qi | Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions[END_REF][START_REF] Qi | On complete monotonicity for several classes of functions related to ratios of gamma functions[END_REF][START_REF] Qi | Lévy-Khintchine representation of Toader-Qi mean[END_REF][START_REF] Qi | The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function[END_REF][START_REF] Qi | Integral representations and properties of some functions involving the logarithmic function[END_REF][START_REF] Qi | Integral representations of bivariate complex geometric mean and their applications[END_REF][START_REF] Qi | Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean[END_REF][START_REF] Qi | The harmonic and geometric means are Bernstein functions[END_REF] and closely related references. This new notion can be used to measure and differentiate complete monotonicity more accurately. For example, the functions 1

x α and 1 x β for α, β > 0 and α = β are both completely monotonic on (0, ∞), but they are different completely monotonic functions. How to quantitatively measure their difference? How to quantitatively differentiate them? The notion of completely monotonic degrees is useful: the completely monotonic degrees of these functions 1

x α and 1 x β for α, β > 0 and α = β are α > 0 and β > 0 respectively. On the other hand, the classical gamma function Γ(z) and polygamma functions ψ (k) for k ≥ 0 and x > 0 are fundamental in mathematics and mathematical sciences. The asymptotic expansions of ln Γ(z) and ψ (k) for k ≥ 0 have extensive applications in almost all mathematical sciences. The remainders (-1) m [R n (x)] (m) for m, n ≥ 0 in [START_REF] Koumandos | Complete monotonicity and related properties of some special functions[END_REF] have been proved to be completely monotonic on (0, ∞) by several mathematicians in Germany, Denmark, Romania, Cyprus, China, and so on. How to quantitatively differentiate complete monotonicity of these remainders? This is the main contribution of this paper.

The functions ln x -ψ(x) and ψ(x) -ln x + 1 2x + 1 12x 2 , which are simpler but applied extensively, and their complete monotonicity on (0, ∞) have been studied for many years by many mathematicians in many top mathematical journals. In this paper, we tried and succeeded to strengthen and differentiate these known conclusions.

What are motivations of advances for pure mathematics? Questions! Problems! Guesses! Conjectures! Hypotheses! Therefore, it is very important to pose correct and significant problems for pure mathematics. To pose correct and significant problems is difficult, even very difficult, because it needs deep and intelligent insight into mathematics. We believe that we posed correct and significant problems in this paper.

Remark 2. This paper is a largely expanded version of the preprint [START_REF] Qi | Notes on complete monotonicity related to the difference of the psi and logarithmic functions[END_REF].

Acknowledgements. The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.


Feng Qi

AND Da-Wei Niu

Dongkyu Lim

NOTES ON EXPLICIT AND INVERSION FORMULAS FOR THE CHEBYSHEV POLYNOMIALS OF THE FIRST TWO KINDS

Keywords: 2010 Mathematics Subject Classification. Primary 11B38, Secondary 11C08, 26C05, 33C45, 33C47, 33D45 explicit formula, inversion formula, Rodrigues formula, Chebyshev polynomial

In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Faà di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.

Introduction

It is well known [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF][START_REF] Mason | Chebyshev Polynomials[END_REF][START_REF]NIST Handbook of Mathematical Functions[END_REF][START_REF] Zwillinger | CRC Standard Mathematical Tables and Formulae[END_REF] that the Chebyshev polynomials of the first and second kinds T n and U n (x) are very important in mathematical sciences and that, in the study of ordinary differential equations [5, pp. xxxv and 1004], they arise as solutions to the Chebyshev differential equations 1 -x 2 y -xy + n 2 y = 0 and 1 -x 2 y -3xy + n(n + 2)y = 0 for the Chebyshev polynomials of the first and second kinds T n and U n respectively.

In [START_REF] Mason | Chebyshev Polynomials[END_REF]Eqs. (4.30) and (4.31)], the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds T n and U n read that

T n (x) = (-1) n 2 n n! (2n)! 1 -x 2 1/2 d n d x n 1 -x 2 n-1/2 (1.1)
and

U n (x) = (-1) n 2 n (n + 1)! (2n + 1)! 1 -x 2 -1/2 d n d x n 1 -x 2 n+1/2 . (1.2)
For variants of the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds T n and U n , please refer to, for example, [5, pp. 1003-1004], [7, p. 442], [START_REF] Qi | Notes on the Rodrigues formulas for two kinds of the Chebyshev polynomials[END_REF]Sectiion 4], and [29, pp. 432-433].

In [5, p. 1003], the Rodrigues formulas for T n (x) and U n (x) are written in the forms

T n (x) = (-1) n √ 1 -x 2 (2n -1)!! d n d x n 1 -x 2 n-1/2 (1.3) and U n (x) = (-1) n (n + 1) √ 1 -x 2 (2n + 1)!! d n d x n 1 -x 2 n+1/2 . (1.4)
In [5, p. 1004] and [29, pp. 432-433], the Rodrigues formulas for T n (x) and U n (x) are formulated as

T n (x) = (-1) n √ π 2 n Γ(n + 1/2) 1 -x 2 1/2 d n d x n 1 -x 2 n-1/2 (1.5)
and

U n (x) = (-1) n √ π (n + 1) 2 n+1 Γ(n + 3/2) 1 -x 2 -1/2 d n d x n 1 -x 2 n+1/2 , (1.6)
where Γ(z) stands for the classical gamma function which can be defined [?, ?] by

Γ(z) = lim n→∞ n!n z n k=0 (z + k) , z ∈ C \ {0, -1, -2, . . . } or by Γ(z) = ∞ 0 t z-1 e -t d t, (z) > 0.
In [7, p. 442], the Rodrigues formulas for T n (x) and U n (x) are arranged as

T n (x) = 1 -x 2 1/2 (-2) n (1/2) n d n d x n 1 -x 2 n-1/2
(1.7) and

U n (x) = (n + 1) 1 -x 2 -1/2 (-2) n (3/2) n d n d x n 1 -x 2 n+1/2 , (1.8)
where (x) n for n ≥ 0 and x ∈ R denotes the rising factorial which can be defined [START_REF] Qi | Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers[END_REF] by

(x) n = n-1 =0 (x + ) = Γ(x + n) Γ(x) = x(x + 1) • • • (x + n -1), n ≥ 1; 1, n = 0.
By virtue of the recurrence relation Γ(x + 1) = xΓ(x), we have

Γ n + 1 2 = n-1 =0 n -- 1 2 Γ 1 2 = 1 2 n √ π = (2n -1)!! 2 n √ π and Γ n + 3 2 = n =0 n -+ 1 2 Γ 1 2 = 3 2 n √ π 2 = (2n + 1)!! 2 n+1 √ π .
Substituting these into (1.5) and (1.6) respectively leads to (1.3), (1.4), (1.7), and (1.8) which are equivalent to (1.1) and (1.2) respectively.

In [29, pp. 432-433], it was listed that

T n (x) = n 2 n/2 m=0 (-1) m (n -m -1)! m!(n -2m)! (2x) n-2m (1.9)
and

U n (x) = n/2 m=0 (-1) m (n -m)! m!(n -2m)! (2x) n-2m , (1.10)
where n ∈ N and t denotes the floor function whose value equals the largest integer less than or equal to t.

In this paper, starting from the four formulas (1.1), (1.2), (1.9), and (1.10), by virtue of the Faà di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem [START_REF] Qi | The inverse of a triangular matrix and several identities of the Catalan numbers[END_REF]Theorem 4.3] for combinatorial coefficients, we will derive the following two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials T n and U n .

Four lemmas

For proving our main results, Theorems 1 and 2 below, we need the following four lemmas.

Lemma 1 ( [4, pp. 134 and 139]). For n ≥ k ≥ 0, the Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind

B n,k (x 1 , x 2 , . . . , x n-k+1 ) by d n d t n f • h(t) = n k=0 f (k) (h(t)) B n,k h (t), h (t), . . . , h (n-k+1) (t) . (2.1) Lemma 2 ( [4, p. 135]). For n ≥ k ≥ 0, we have B n,k abx 1 , ab 2 x 2 , . . . , ab n-k+1 x n-k+1 = a k b n B n,k (x 1 , x 2 , . . . , x n-k+1 ), (2.2)
where a and b are any complex numbers.

Lemma 3 ( [13, Theorem 5.1] and [25, Section 3]). For n ≥ k ≥ 0, the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n-k+1 ) satisfy

B n,k (x, 1, 0, . . . , 0) = 1 2 n-k n! k! k n -k x 2k-n , (2.3)
where 0 0 = 1 and p q = 0 for q > p ≥ 0. Lemma 4 ( [26, Theorem 4.3]). For n ≥ k ≥ 1, let s k and S k be two sequences independent of n. Then

s n n! = n k=1 (-1) k k n -k S k
if and only if

nS n = n k=1 (-1) k (k -1)! 2n -k -1 n -1 s k .

Main results and their proofs

Now we begin to state and prove our main results, Theorems 1 and 2 below.

Theorem 1. For n ≥ 0, the Chebyshev polynomials T n and U n can be explicitly computed by

T n (x) = x n n/2 =0 n 2 1 - 1 x 2 (3.1)
and

U n (x) = x n n/2 =0 n + 1 2 + 1 1 - 1 x 2 . (3.2)
Proof. By virtue of the formuals (2.1), (2.2), and (2.3), we have

d n d x n 1 -x 2 n-1/2 = n k=1 d k u n-1/2 d u k B n,k (-2x, -2, 0 . . . , 0) = n k=1 k-1 =0 n -- 1 2 u n-k-1/2 (-2) k B n,k (x, 1, 0 . . . , 0) = n k=1 1 2 k k-1 =0 (2n -2 -1) 1 -x 2 n-k-1/2 (-2) k 1 2 n-k n! k! k n -k x 2k-n = n! (2x) n 1 -x 2 n-1/2 n k=1 (-1) k k n -k (2n -1)!! [2(n -k) -1]!! 2 k k! x 2 1 -x 2 k = n!(2n -1)!! (2x) n 1 -x 2 n-1/2 n k=1 (-1) k k n -k 2 k k![2(n -k) -1]!! x 2 1 -x 2 k , where n ∈ N, u = u(x) = 1 -x 2 ,
and the double factorial of negative odd integers -2n -1 is defined by

(-2n -1)!! = (-1) n (2n -1)!! = (-1) n 2 n n! (2n)! , n ≥ 0.
Substituting the above established equality into (1.1) and simplifying lead to

T n (x) = n k=1 (-1) n-k 4 n-k k n -k n k [2(n -k)]!! [2(n -k) -1]!! x 2k-n 1 -x 2 n-k
which can be rearranged, by replacing n -k by , as

T n (x) = x n n-1 =0 (-1) 4 n n - n - (2 )!! (2 -1)!! 1 x 2 -1 . Since 1 4 n n - n - (2 )!! (2 -1)!! = n 2 ,
we arrives at the identity (3.1).

Repeating the above process, we can obtain

d n d x n 1 -x 2 n+1/2 = n k=1 d k u n+1/2 d u k B n,k (-2x, -2, 0 . . . , 0) = n k=1 k-1 =0 n -+ 1 2 u n-k+1/2 (-2) k B n,k (x, 1, 0 . . . , 0) = n k=1 1 2 k k-1 =0 (2n -2 + 1) 1 -x 2 n-k+1/2 (-2) k 1 2 n-k n! k! k n -k x 2k-n = n! (2x) n 1 -x 2 n+1/2 n k=1 (-1) k k n -k (2n + 1)!! [2(n -k) + 1]!! 2 k k! x 2 1 -x 2 k = n!(2n + 1)!! (2x) n 1 -x 2 n+1/2 n k=1 (-1) k k n -k 2 k k![2(n -k) + 1]!! x 2 1 -x 2 k .
Substituting this into (1.2) and simplifying lead to

U n (x) = n k=1 (-1) n-k 2 2n-2k+1 k n -k n + 1 k [2(n -k + 1)]!! [2(n -k) + 1]!! x 2k-n 1 -x 2 n-k .
Replacing n -k by reveals that

U n (x) = n-1 =0 (-1) 2 2 +1 n + 1 n - n -[2( + 1)]!! (2 + 1)!! x n-2 1 -x 2 .
Due to

1 2 2 +1 n + 1 n - n -[2( + 1)]!! (2 + 1)!! = n + 1 2 + 1 ,
we derive (3.2). The proof of Theorem 1 is complete.

Theorem 2. For n ∈ N, we have

n k=1 2n -k -1 n -1 (2x) k T k (x) = 1 2 (2x) 2n (3.3)
and

n k=1 k 2n -k -1 n -1 (2x) k U k (x) = n(2x) 2n . (3.4)
Proof. We notice that the formulas (1.9) and (1.10) can be rearranged as

T n (x) = n 2 n/2 m=0 (-1) m n -m m (2x) n-2m n -m (3.5)
and

U n (x) = n/2 m=0 (-1) m n -m m (2x) n-2m . (3.6)
The inversion theorem in Lemma 4 can be restated as

(-1) n s n n! = n-1 =0 (-1) n -S n-= n/2 =0 (-1) n -S n-
if and only if

nS n = n =1 (-1) ( -1)! 2n --1 n -1 s .
The formulas (3.5) and (3.6) can be rearranged as

2 n (2x) n T n (x) = n/2 =0 (-1) n -(2x) 2(n-) n - and (2x) n U n (x) = n/2 =0 (-1) n -(2x) 2(n-) .
Consequently, we obtain

n (2x) 2n n = n k=1 (-1) k (k -1)! 2n -k -1 n -1 (-1) k 2(k -1)!(2x) k T k (x)
and

n(2x) 2n = n k=1 (-1) k (k -1)! 2n -k -1 n -1 (-1) k k!(2x) k U k (x)
which can be simplified as (3.3) and (3.4). The proof of Theorem 2 is complete.

Remarks

In this section, we will list several remarks to explain more about the formula (2.3), Lemma 4, our main results, and other things.

Remark 1. To the best of our knowledge, the nice formula (2.3) was first concluded in [START_REF] Qi | Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials[END_REF] and has been extensively applied in the papers [8-16, 18, 19, 21, 23, 25, 27, 28] and closely related references therein. The formula (2.3) has been generalized in the papers [START_REF] Qi | Viewing some ordinary differential equations from the angle of derivative polynomials[END_REF][START_REF] Qi | Explicit formulas and identities on Bell polynomials and falling factorials[END_REF][START_REF] Qi | Special values of the Bell polynomials of the second kind for some sequences and functions[END_REF] and closely related references therein.

Remark 2. To the best of our knowledge, Lemma 4 is a new inversion theorem and has been applied in the paper [START_REF] Qi | A simple form for coefficients in a family of ordinary differential equations related to the generating function of the Legendre polynomials[END_REF][START_REF] Qi | Some identities for a sequence of unnamed polynomials connected with the Bell polynomials[END_REF][START_REF] Qi | Notes on the Rodrigues formulas for two kinds of the Chebyshev polynomials[END_REF][START_REF] Qi | Simplifying coefficients in differential equations for generating function of Catalan numbers[END_REF]. Remark 3. Because both the formula (2.3) and Lemma 4 are new, our main results stated in Theorems 1 and 2, or at least their proofs, are also new.

Remark 4. The Chebyshev polynomials are classical, but their study is still very active. As examples, we recommend three newly-published papers [START_REF] Cesarano | Integral representations and new generating functions of Chebyshev polynomials[END_REF][START_REF] Cesarano | Multi-dimensional Chebyshev polynomials: a non-conventional approach[END_REF][START_REF] Cesarano | A note on two-variable Chebyshev polynomials[END_REF] to readers. Considering the length of this paper, we would not like to detail main results in these three papers and the closely-related references therein.

Remark 5. This paper is a slightly revised version of the preprint [START_REF] Qi | Notes on the Rodrigues formulas for two kinds of the Chebyshev polynomials[END_REF].

Acknowledgements. The third author was supported by the National Research Foundation of Korea (Grant No. 2018R1D1A1B07041846).

The authors appreciate anonymous referees for their valuable comments on the original version of this paper.

Authors' addresses


Kottakkaran Sooppy Nisar

Feng Qi

Gauhar Rahman

Shahid Mubeen

Muhammad Arshad

Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function [1]

Keywords: extended gamma function, inequality, Kummer confluent hypergeometric k-function, logarithmic convexity AMS Subject Classification: Primary 33B15, Secondary 33C15, 33B20, 33C05, 41A60 [1]

published or not. The documents may come

Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric

k-function

Introduction

The gamma function Γ can be defined [START_REF] Qi | Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function[END_REF][START_REF] Rainville | Special Functions[END_REF][START_REF] Yang | Monotonicity and inequalities for the gamma function[END_REF][START_REF] Yang | Monotonicity and sharp inequalities related to gamma function[END_REF] by

Γ(z) = ∞ 0 t z-1 e -t d t, (z) > 0.
Alternatively, it can also be defined [START_REF] Qi | Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities[END_REF] by

Γ(z) = lim n→∞ n!n z-1 (z) n ,
where (z) n for z = 0 is the Pochhammer symbol defined [START_REF] Qi | Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers[END_REF] as

(z) n =    z(z + 1)(z + 2) • • • (z + n -1), n ≥ 1; 1, n = 0.
The relation between (z) n and Γ(z) is

(z) n = Γ(z + n) Γ(z) .
The beta function B(x, y) can be defined [START_REF]NIST Handbook of Mathematical Functions[END_REF][START_REF] Qi | Parametric integrals, the Catalan numbers, and the beta function[END_REF][START_REF] Qi | Catalan numbers, k-gamma and k-beta functions, and parametric integrals[END_REF] by B(x, y) = 1 0 t x-1 (1 -t) y-1 d t, (x), (y) > 0 and can be expressed by B(x, y) = Γ(x)Γ(y) Γ(x + y) , (x), (y) > 0.

In 1995, Chaudhry and Zubair [START_REF] Chaudhry | On the decomposition of generalized incomplete gamma functions with applications of Fourier transforms[END_REF] introduced the extended gamma function

Γ b (z) = ∞ 0 t z-1 e -t-bt -1 d t, (z) > 0, b ≥ 0. (1.1)
If b = 0, then Γ b becomes the classical gamma function Γ.

In 1997, Chaudhry et al [START_REF] Chaudhry | Extension of Euler's beta function[END_REF] introduced the extended beta function

B b (x, y) = 1 0 t x-1 (1 -t) y-1 e -b/t(1-t) d t, (b), (x)
, (y) > 0.

(

It is clear that B 0 (x, y) = B(x, y).

In 2009, Barnard et al [START_REF] Barnard | A note on Turán type and mean inequalities for the Kummer function[END_REF] established three inequalities [φ(a, a + b, x)] 2 > φ(a + v, a + b, x)φ(a -v, a + b, x),

[φ(a, c, x)] 2 > φ(a + v, c, x)φ(a -v, c, x), and

A(φ(a + v, a + b, x), φ(a -v, a + b, x)) > φ(a, a + b, x) > G(φ(a + v, a + b, x), φ(a -v, a + b, x)),
where A(α, β) = α+β 2 and G(α, β) = √ αβ are the arithmetic and geometric means and

φ(a, b, x) = ∞ n=0 (a) n (b) n x n n!
is the Kummer confluent hypergeometric function [START_REF] Qi | On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function[END_REF][START_REF] Rainville | Special Functions[END_REF].

The Kummer confluent hypergeometric k-function is defined by

φ k (a, b, x) = ∞ n=0 (a) n,k (b) n,k x n n! , where (a) n,k = a(a + k)(a + 2k) • • • [a + (n -1)k]
for n ≥ 1 and k > 0 with (a) 0,k = 1 is the Pochhammer k-symbol, which can also be rewritten as

(a) n,k = Γ k (a + nk) Γ k (a)
and the gamma k-function Γ k (a) is defined [START_REF] Díaz | On hypergeometric functions and Pochhammer k-symbol[END_REF][START_REF] Qi | Parametric integrals, the Catalan numbers, and the beta function[END_REF][START_REF] Qi | Catalan numbers, k-gamma and k-beta functions, and parametric integrals[END_REF] by

Γ k (a) = ∞ 0 t a-1 e -t k /k d t
In 2012, Mubeen [START_REF] Mubeen | k-analogue of Kummer's first formula[END_REF] introduced the k-analogue of Kummer's transformation as

φ k (a, b, x) = e x φ k (a, b -a, -x). (1.3)
In Section 2, we prepare two lemmas. In Section 3, we discuss applications of some integral inequalities such as the Chebychev integral inequality. In Section 4, we prove the logarithmic convexity of the extended gamma function. In the last section, we introduce a mean inequality of Turán type for the Kummer confluent hypergeometric k-function.

Lemmas

In order to obtain our main results, we need the following lemmas.

Lemma 2.1 (Chebychev integral inequality [START_REF] Dragomir | Inequalities for beta and gamma functions via some classical and new integral inequalities[END_REF][START_REF] Dragomir | Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules[END_REF][START_REF] Kumar | Some inequalities involving beta and gamma functions[END_REF][START_REF] Qi | Some inequalities constructed by Tchebysheff's integral inequality[END_REF]) Let f, g, h : I ⊆ R → R be mappings such that h(x) ≥ 0, h(x)f (x)g(x), h(x)f (x), and h(x)g(x) are integrable on I. If f (x) and g(x) are synchronous (or asynchronous, respectively) on I, that is,

[f (x) -f (y)][g(x) -g(y)] 0 for all x, y ∈ I, then I h(x) d x I h(x)f (x)g(x) d x I h(x)f (x) d x I h(x)g(x) d x.
Lemma 2.2 (Hölder's inequality [START_REF] Rudin | Real and complex analysis[END_REF][START_REF] Tian | Properties of generalized sharp Hölder's inequalities[END_REF]) Let p and q be positive real numbers such that 1 p + 1 q = 1 and f, g : [c, d] → R be integrable functions. Then

d c f (x)g(x) d x ≤ d c |f (x)| p d x 1/p d c |g(z)| q d x 1/q .
3 Inequalities involving the extended gamma function via the Chebychev integral inequality

In this section, we prove some inequalities involving the extended gamma function via the Chebychev integral inequality in Lemma 2.1.

Theorem 3.1 Let m, p and r be positive real numbers such that p > r > -m. If r(p -m -r) 0, then

Γ b (m)Γ b (p) Γ b (p -r)Γ b (m + r). (3.1)
Proof Let us define the mappings f, g, h : [0, ∞) → [0, ∞) given by f (t) = t p-r-m , g(t) = t r , and h(t) = t m-1 e -t-bt -1 .

If r(p -m -r) 0, then we can claim that the mappings f and g are synchronous (asynchronous) on (0, ∞). Thus, by applying the Chebychev inequality on I = (0, ∞) to the functions f , g and h defined above, we can write

∞ 0 t m-1 e -t-bt -1 d t ∞ 0 t p-r-m t r t m-1 e -t-bt -1 d t ∞ 0 t p-r-m t m-1 e -t-bt -1 d t ∞ 0 t r t m-1 e -t-bt -1 d t.
This implies that

∞ 0 t m-1 e -t-bt -1 d t ∞ 0 t p-1 e -t-bt -1 d t ∞ 0 t p-r-1 e -t-bt -1 d t ∞ 0 t m+r-1 e -t-bt -1 d t.
By (1.1), we acquire the required inequality (3.1).

Corollary 3.1 If p > 0 and q ∈ R with |q| < p, then

Γ b (p) ≤ [Γ b (p -q)Γ b (p + q)] 1/2 .
Proof By setting m = p and r = q in Theorem 3.1, we obtain r(p-m-r) = -q 2 ≤ 0 and then the inequality (3.1) provides the desired Corollary 3.1.

Theorem 3.2 If m, n > 0 are similarly (oppositely) unitary, then

Γ b (m + n + b) Γ b (m + b + 1)Γ b (n + b + 1) Γ b (b + 2) . Proof Consider the mappings f, g, h : [0, ∞) → [0, ∞) defined by f (t) = t m-1 , g(t) = t n-1 , and h(t) = t b+1 e -t-bt -1 .
Now if the condition (m-1)(n-1) 0 holds, then the Chebychev integral inequality applied to the functions f , g, and h means

∞ 0 t b+1 e -t-bt -1 d t ∞ 0 t m-1 t n-1 t b+1 e -t-bt -1 d t ∞ 0 t m-1 t b+1 e -t-bt -1 d t ∞ 0 t n-1 t b+1 e -t-bt -1 d t.
This implies that

∞ 0 t b+1 e -t-bt -1 d t ∞ 0 t m+n+b-1 e -t-bt -1 d t ∞ 0 t m+b e -t-bt -1 d t ∞ 0 t n+b e -t-bt -1 d t.
By the definition of the extended gamma function, we have

Γ b (b + 2)Γ b (m + n + b) Γ b (m + b + 1)Γ b (n + b + 1), or Γ b (m + n + b) Γ b (m + b + 1)Γ b (n + b + 1) Γ b (b + 2) .
The required proof is complete. Proof Consider the mappings f, g, h : [0, ∞) → [0, ∞) defined by

Corollary 3.2 If b = 0, then Γ(m + n) mnΓ(m)Γ(n).
f (t) = t m , g(t) = t n , and h(t) = t b e -t-bt -1 .
If the conditions of Theorem 3.1 hold, then the mappings f and g are synchronous (asynchronous) on [0, ∞). Thus, by applying the Chebychev integral inequality in Lemma 2.1 to the functions f, g and h defined above, we have

∞ 0 t b e -t-bt -1 d t ∞ 0 t m t n t b e -t-bt -1 d t ∞ 0 t m t b e -t-bt -1 d t ∞ 0 t n t b e -t-bt -1 d t.
This implies that

∞ 0 t b e -t-bt -1 d t ∞ 0 t m+n+b e -t-bt -1 d t ∞ 0 t m+b e -t-bt -1 d t ∞ 0 t n+b e -t-bt -1 d t.
Thus by the definition of the extended gamma function, we have

Γ b (b + 1)Γ b (m + n + b + 1) Γ b (m + b + 1)Γ b (n + b + 1).
The required proof is complete.

Corollary 3.3 If b = 0, then Γ(m + n) mnΓ(m)Γ(n) m + n .
4 Log-convexity of the extended gamma function

It is well known that, if f > 0 and ln f is convex, then f is said to be a logarithmically convex function. Every logarithmically convex must be convex. See [START_REF] Nantomah | The k-analogues of some inequalities for the digamma function[END_REF] and [18, Remark 1.9]. In this section, we verify the log-convexity of the extended gamma function.

Theorem 4.1 The extended gamma function Γ b : (0, ∞) → R is logarithmically convex.

Proof Let p and q be positive numbers such that

1 p + 1 q = 1. Since Γ b x p + y q ≤ [Γ b (x)] 1/p [Γ b (y)] 1/q , see [4], letting λ = 1 p and (1 -λ) = 1 q leads to Γ b [λx + (1 -λ)y] ≤ [Γ b (x)] λ [Γ b (y)] (1-λ) .
As a result, the function Γ b is logarithmically convex.

A mean inequality of the Turán type for the Kummer confluent hypergeometric k-function

In this section, we present a mean inequality involving the Kummer confluent hypergeometric k-function. For this purpose, we consider the relation

φ k (a + k, b, x) -φ k (a, b, x) = kx b φ k (a + k, b + k, x), k > 0.
(5.1)

Theorem 5.1 For a, b, k > 0 and v ∈ N with a, b ≥ v -k, the inequality [φ k (a, a + b, x)] 2 > φ k (a + v, a + b, x)φ k (a -v, a + b, x) (5.2)
is valid for all nonzero x ∈ R.

First proof Assume that x > 0. For c = 0, -1, -2, . . . , define

f v,k (x) = [φ k (a, c, x)] 2 -φ k (a + v, c, x)φ k (a -v, a + b, x)
and

f v+k,k (x) = φ k (a, c, x) 2 -φ k (a + v + k, c, x)φ k (a -v -k, a + b, x).
From (5.1), it follows that

f v+k,k (x) -f v,k (x) = φ k (a + v, c, x)φ k (a -v, c, x) -φ k (a + v + k, c, x)φ k (a -v -k, c, x) = φ k (a -v, c, x)[φ k (a + v, c, x) -φ k (a + v + k, c, x)] + φ k (a + v + k, c, x)[φ k (a -v, c, x) -φ k (a -v -k, c, x)] = φ k (a -v, c, x) -kx c φ k (a + v + k, c + k, x) + φ k (a + v + k, c, x) kx c φ k (a -v, c + k, x) = kx c g v,k (x),
where

g v,k (x) = φ k (a + v + k, c, x)φ k (a -v, c + k, x) -φ k (a -v, c, x)φ k (a + v + k, c + k, x).
Accordingly, by the Cauchy product, we have

g v,k (x) = ∞ s=0 s r=0 (a + v + k) s,k (a -v) s-r,k r!(s -r)! × 1 (c) s,k (c + k) s-r,k - 1 (c) s-r,k (c + k) s,k x s = ∞ s=0 s r=0 (a + v + k) s,k (a -v) s-r,k r!(s -r)! (c + mk) -(c + nk -mk) c(c + k) s,k (c + k) s-r,k x s = k c ∞ s=0 s r=0 T s,r,k (2r -s)x s ,
where

T s,r,k = (a + v + k) s,k (a -v) s-r,k r!(s -r)!(c + k) s,k (c + k) s-r,k .
If s is even, then

s r=0 T s,r,k (2r -s) = s/2-1 r=0 T s,r,k (2r -s) + s r=s/2+1 T s,r,k (2r -s) = s/2-1 r=0 T s,r,k (2r -s) + s/2-1 r=0 T s,s-r,k (2(s -r) -s) = (s-1)/2 r=0 (T s,s-r,k -T s,r,k )(s -2r),
where x denotes the ceiling function whose value is the greatest integer not more than x. Similarly, if s is odd,

s r=0 T s,r,k (2r -s) = (s-1)/2 r=0 (T s,s-r,k -T s,r,k )(s -2r).
Accordingly,

f v+k,k (x) -f v,k (x) = kx c g v,k (x) = k 2 x c 2 ∞ s=1 (s-1)/2 r=0 (T s,s-r,k -T s,r,k )(s -2r)x s . (5.3)
Carefully simplifying gives

T s,s-r,k -T s,r,k = (a + v + k) s,k (a -v) s,k -(a + v + 1) s,k (a -v) s,k r!(s -r)!(c + k) s-r,k (c + k) s,k = (a + v + k) s,k (a -v) s,k r!(s -r)!(c + k) s-r,k (c + k) s,k (a + v + k) s-r,k (a + v + k) s,k - (a -v) s-r,k (a -v) s,k = (a + v + k) s,k (a -v) s,k r!(s -r)!(c + k) s-r,k (c + k) s,k [h k (a + v + k) -h k (a -v)], (5.4)
where

h k (x) = (x) s-r,k (x) s,k . For x > 0 and s -r > r, that is, [ s-1 2 ] ≥ r, the logarithmic derivatives of h k is h k (x) h k (x) = ψ k (x + (s -r)k) -ψ k (x + nk) > 0,
where [START_REF] Díaz | On hypergeometric functions and Pochhammer k-symbol[END_REF][START_REF] Kokologiannaki | Properties and inequalities of generalized k-gamma, beta and zeta functions[END_REF][START_REF] Nantomah | The k-analogues of some inequalities for the digamma function[END_REF]). Hence, the function h k is increasing under the condition stated. This fact together with the aid of (5.3) and (5.4) yields

ψ k = Γ k Γ k is the digamma k-function (see
f v+k,k (x) -f v,k (x) = kx c g v,k (x) = k 2 x c 2 ∞ s=1 (s-1)/2 r=0 (T s,s-r,k -T s,r,k )(s -2r)x s > 0, (5.5)
where a ≥ v ≥ 0, x > 0, c + k > 0, and c = 0. Consequently, from (5.5), it follows that

f v+k,k (x) = [f v+k,k (x) -f v,k (x)] + [f v,k (x) -f v-k,k (x)] + • • • + [f 1,k (x) -f 0,k (x)] is positive for a ≥ v ≥ v -k ≥ v -2k ≥ • • • ≥ 0 and f 0,k (x) = 0. Now replacing v by v -k shows that f v,k (x) > 0, x > 0, v ∈ N, a ≥ v -k. (5.6)
Therefore, the function f v,k is absolutely monotonic on (0, ∞), that is, f

( ) v,k (x)
> 0 for = 0, 1, 2, . . . . This proves Theorem 5.1 for the case x > 0. Now suppose that x < 0, a, b > 0, and v ∈ N with a, b ≥ v -k. Since φ k (a, c, x) is symmetric in a and b, by interchanging a and b in Theorem 5.1, we obtain

φ k (b, a + b, -x) 2 -φ k (b + v, a + b, -x)φ k (b -v, a + b, -x) > 0.
By using Kummer's transformation (1.3), we have

e -2x φ k (a, a + b, x) 2 -e -2x φ k (a -v, a + b, x)φ k (a + v, a + b, x) > 0.
Thus, Theorem 5.1 also holds for x < 0.

Second proof Since (a) n,k = a(a + k)(a + 2k) • • • (a + (n -1)k) = k n a k a k + 1 a k + 2 • • • a k + (n -1) = k n a k n , it follows that φ k (a, b; x) = ∞ n=0 (a) n,k (b) n,k x n n! = ∞ n=0 k n (a/k) n k n (b/k) n x n n! = φ a k , b k , x .
[φ k (a, c, x)] 2 ≥ φ k (a -v, c, x)φ k (a + v, c, x) holds for any v ∈ N with a ≥ v -k.
Proof This follows directly from the proof of Theorem 5.1 and the fact that the equation (5.6) holds under the conditions c + k > 0 and c = 0.

Corollary 5.2 If v ∈ N and a, b ≥ v, then A φ k (a + v, a + b, x), φ k (a -v, a + b, x) > φ k (a, a + b, x) > G φ k (a + v, a + b, x), φ k (a -v, a + b, x) (5.7)
for all nonzero x ∈ R, where A and G are, respectively, the arithmetic and geometric means.

Proof First assume x ≥ 0 and a, b ≥ v for v ∈ N. Then the left hand side inequality in (5.7) is a direct consequence of the facts that

A((a + v) s,k , (a -v) s,k ) = (a) s,k
for s = 0, 1 and

A((a + v) s,k , (a -v) s,k ) > (a) s,k
for s ≥ 2. Hence, by induction, we have

A(φ k (a + v, a + b, x), φ k (a -v, a + b, x)) = ∞ s=0 A((a + v) s,k , (a -v) s,k )x s (a + b) s,k s! > ∞ s=0 (a) s,k x s (a + b) s,k s! = φ k (a, a + b, x).
For x ≥ 0, the right hand side inequality in (5.7) follows from taking square root of (5.2). The proof of Corollary 5.2 for x ≥ 0 is thus complete. Now assume x < 0 with a, b ≥ v. Interchanging a and b in (5.7) one arrives at

A(φ k (b + v, a + b, -x), φ k (b -v, a + b, -x)) > φ k (b, a + b, -x) > G(φ k (b + v, a + b, -x), φ k (b -v, a + b, -x)).
Making use of the k-analogue of Kummer's transformation and the homogeneity of A and G acquires

e -x A(φ k (a -v, a + b, x), φ k (a + v, a + b, x)) > e -x φ k (a, a + b, x) > e -x G(φ k (a -v, a + b, x), φ k (a + v, a + b, x)).
Consequently, Theorem 5.7 also follows for x < 0. Remark 5.2 In [START_REF] Qi | Some inequalities of the Turán type for confluent hypergeometric functions of the second kind[END_REF], some inequalities of the Turán type for confluent hypergeometric functions of the second kind were also discovered.

Remark 5.3 By the way, we note that the papers [START_REF] Guo | Monotonicity of functions connected with the gamma function and the volume of the unit ball[END_REF][START_REF] Ivády | A note on a gamma function inequality[END_REF][START_REF] Kupán | A result regarding monotonicity of the Gamma function[END_REF][START_REF] Kupán | Monotonicity theorems and inequalities for the gamma function[END_REF][START_REF] Qi | Integral representations and properties of some functions involving the logarithmic function[END_REF][START_REF] Yang | Monotonicity and sharp inequalities related to gamma function[END_REF][START_REF] Zhao | A refinement of a double inequality for the gamma function[END_REF] belong to the same series in which inequalities and complete monotonicity for functions involving the gamma function Γ(x) and the logarithmic function ln(1 + x) were discussed.

Remark 5.4 This paper is a slightly revised version of the preprint [START_REF] Nisar | Some inequalities involving the extended gamma and beta functions[END_REF].

Conclusions

In this paper, we present some inequalities involving the extended gamma function Γ b (z) via some classical inequalities such as the Chebychev inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the logconvexity of the extended gamma function Γ b (z) by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function φ(z).

Theorem 3 . 3

 33 If m and n are positive real numbers such that m and n are similarly (oppositely) unitary, then Γ b (b + 1)Γ b (m + n + b + 1) Γ b ((m + b + 1)Γ b (n + b + 1), b ≥ 0.

1 . 5 . 1

 151 Replacing a and b by a k and b k , respectively, gives Theorem 5.Corollary If a > 0 and c + k > 0 with c = 0, then the inequality

Remark 5 . 1

 51 In Section 5, we have established a Turán type and mean inequality for k-analogue of the Kummer confluent hypergeometric function. If we let k → 1, then we can conclude to the corresponding inequalities of the confluent hypergeometric function.

Acknowledgement

The authors appreciate the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.