Feng Qi

Ravi Bhukya

Venkatalakshmi Akavaram
email: akavaramvlr@gmail.com

Some inequalities of the Turán type for confluent hypergeometric functions of the second kind

Keywords: Mathematics Subject Classification (2010): Primary 26D15, Secondary 26D20, 26D99, 33C15, 44A10, 44A15 inequality of the Turán type, confluent hypergeometric function of the second kind, improvement, Mellin transform, Laplace transform, Hölder integral inequality

In the paper, by virtue of the Hölder integral inequality, the authors derive some inequalities of the Turán type for confluent hypergeometric functions of the second kind, for the Mellin transforms, and for the Laplace transforms, and improve some known inequalities of the Turán type.

Introduction

In 1950, P. Turán [START_REF] Turán | On the zeros of the polynomials of Legendre[END_REF] proved that the Legendre polynomials P n (x) satisfy P n (x)P n+2 (x) -P 2 n+1 (x) ≤ 0 for |x| ≤ 1 and n = 0, 1, 2, . . . , where the equality holds only if x = ±1. An inequality of this kind is known as an inequality of the Turán type. This classical inequality has been extended to various special functions. For recent development on this classical inequality, please refer to [START_REF] Baricz | Turán type inequalities for hypergeometric functions[END_REF][START_REF] Baricz | Turán type inequalities for modified Bessel functions[END_REF][START_REF] Baricz | Turán type inequalities for Tricomi confluent hypergeometric functions[END_REF][START_REF] Baricz | On Turán type inequalities for modified Bessel functions[END_REF][START_REF] Baricz | Turán type inequalities for confluent hypergeometric function of second kind[END_REF][START_REF] Laforgia | Turán type inequalities for some special functions[END_REF] and closely related reference therein.

It is known [1, p. 504-505] that confluent hypergeometric functions of the second kind ψ(a, c, x) are also known as the Tricomi confluent hypergeometric functions, are a special solution of Kummer's differential equation xy (x) + (c -x)y (x) -ay(x) = 0, and have the integral representation

ψ(a, c, x) = 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -xt d t (1.1)
for a > 0, c ∈ R, and x > 0, where

Γ(z) = ∞ 0 t z-1 e -t d t, (z) > 0
is the classical Euler gamma function [START_REF] Qi | Inequalities of the Grünbaum type for completely monotonic functions[END_REF][START_REF] Qi | Complete monotonicity of the divided differences of di-and tri-gamma functions and its applications[END_REF][START_REF] Qi | Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function[END_REF][START_REF] Qi | A logarithmically completely monotonic function involving the ratio of gamma functions[END_REF]. The Laplace transform and the Mellin transform of a function f (t) are respectively defined by

L(s) = L (f )(s) = ∞ 0 f (t)e -st d t, s > 0 and M (s) = M (f )(s) = ∞ 0 f (t)t s-1 d t.
These transforms are widely-used integral transforms with many applications in physics and engineering.

In this paper, we will study some Turán type inequalities for confluent hypergeometric functions of the second kind ψ(a, c, x).

Inequalities of the Turán type

We are now in a position to find some inequality of the Turán type for confluent hypergeometric functions of the second kind. These newly-founded inequalities improve existed inequality of the Turán type in [START_REF] Baricz | Turán type inequalities for Tricomi confluent hypergeometric functions[END_REF]Theorem 2].

Theorem 2.1. For x > 0, a > 0, and c ∈ R, we have

ψ 2 (a + 1, c, x) < a + 1 a ψ(a, c, x)ψ(a + 2, c, x). (2.1)
Proof. The equality (1.1) can be reformulated as

f (a) ψ(a, c, x)Γ(a) = ∞ 0 t 1 + t a (1 + t) c-1 e -xt 1 t d t.
Replacing a by αp + (1 -α)q for p, q > 0, p = q, and α ∈ (0, 1) and using the well-known Hölder integral inequality give

f (αp + (1 -α)q) = ∞ 0 t 1 + t αp+(1-α)q (1 + t) c-1 e -xt 1 t d t = ∞ 0 t p-1 (1 + t) c-p-1 e -xt α t q-1 (1 + t) c-q-1 e -xt 1-α < ∞ 0 t p-1 (1 + t) c-p-1 e -xt α ∞ 0 t q-1 (1 + t) c-q-1 e -xt 1-α = f α (p)f 1-α (q).
This implies that the function f is strictly logarithmically convex on (0, ∞). Consequently, taking α = 1 2 , p = a, and q = a + 2 leads to f 2 (a + 1) < f (a)f (a + 2) which is equivalent to (2.1). The proof of Theorem 2.1 is complete.

Theorem 2.2. For x > 0, 0 < a 1 , a 2 < a, and c ∈ R, we have

ψ 2 (a, c, x) < Γ(a 1 )Γ(a 2 ) Γ 2 (a) ψ(a 1 , c, x)ψ(a 2 , c, x). (2.2)
Proof. We continue to adopt the notation f (a) in the proof of Theorem 2.1. Then

f (a) = d d a ∞ 0 t 1 + t a (1 + t) c-1 1 t e -xt d t = ∞ 0 t 1 + t a ln t 1 + t (1 + t) c-1 1 t e -xt d t < 0.
Since ln t 1+t < 0 for t > 0, the function f (a) is decreasing on (0, ∞) with respect to a. Accordingly, for 0 < a 1 , a 2 < a, we have f (a) < f (a 1 ) and f (a) < f (a 2 ). Consequently, it follows that f 2 (a) < f (a 1 )f (a 2 ) which is equivalent to (2.2). The proof of Theorem 2.2 is complete. Theorem 2.3. For x > 0, a > 0, and c 1 , c 2 < c ∈ R, we have

ψ(a, c, x)ψ(a, c -1, x) < ψ 2 (a, c + 1, x) < ψ(a, c, x)ψ(a, c + 2, x) (2.3)
and ψ(a, c 1 , x)ψ(a, c 2 , x) < ψ 2 (a, c, x).

(2.4)

Proof. A straightforward computation yields

ψ (c) = 1 Γ(a) d d c ∞ 0 t a-1 (1 + t) c-a-1 e -xt d t = 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 ln(1 + t)e -xt d t > 0.
This means that the function ψ(a, c, x) is increasing with respect to c ∈ R. Hence, for c 1 , c 2 < c, it follows that ψ(a, c 1 , x) < ψ(a, c, x) and ψ(a, c 2 , x) < ψ(a, c, x). Consequently, we obtain the inequality (2.4).

Taking c 1 = c -1 and c 2 = c and replacing c by c + 1 in (2.4) deduce that ψ(a, c -1, x)ψ(a, c, x) < ψ 2 (a, c + 1, x).

From this inequality and the inequality ψ 2 (a, c + 1, x) < ψ(a, c, x)ψ(a, c + 2, x) in the paper [START_REF] Baricz | Turán type inequalities for Tricomi confluent hypergeometric functions[END_REF], the inequality (2.3) follows immediately. The proof of Theorem 2.3 is complete.

Theorem 2.4. For x, y > 0, a > 0, p, q > 0 such that 1 p + 1 q = 1, and c ∈ R, we have ψ a, c, x p + y q < ψ 1/p (a, c, x)ψ 1/q (a, c, y).

(2.5)

Proof. Applying the well-known Hölder integral inequality to the third variable x in ψ(a, c, x) arrives at

ψ a, c, x p + y q = 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -(x/p+y/q)t d t = 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -xt 1/p t a-1 (1 + t) c-a-1 e -yt 1/q < 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -xt 1/p 1 Γ(a) ∞ 0 t q-1 (1 + t) c-q-1 e -yt 1/q = ψ 1/p (a, c, x)ψ 1/q (a, c, x).
Therefore, the inequality (2.5) is proved. The proof of Theorem 2.4 is complete.

Theorem 2.5. For x, y > 1 such that 1

x + 1 y ≤ 1, a > 0, p, q > 0 such that 1 p + 1 q = 1, and c ∈ R, we have ψ a, c, x p p + y q q < ψ 1/p (a, c, px)ψ 1/q (a, c, qy).

(

Proof. Applying Young's inequality to the third variable x in ψ(a, c, x) results in

ψ x p p + y q q = 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -(x p /p+y q /q)t d t ≤ 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -xyt ≤ 1 Γ(a) ∞ 0 t a-1 (1 + t) c-a-1 e -pxt 1/p × 1 Γ(a) ∞ 0 t q-1 (1 + t) c-q-1 e -qyt 1/q = ψ 1/p (a, c, px)ψ 1/q (a, c, qx).
The inequality (2.6) is thus proved. The proof of Theorem 2.5 is complete.

Theorem 2.6. For x, y > 0, a > 0, and c ∈ R, we have ψ 2 (a, c, x + y) < ψ(a, c, x)ψ(a, c, y).

(2.7)

For x > 0, 0 < y < 1, a > 0, and c ∈ R, we have ψ(a, c, x + y) < ψ(a, c, xy).

(2.8)

Proof. It is easy to see that the function ψ(a, c, x) is decreasing with respect to x ∈ (0, ∞). Since x < x + y and y < x + y, it follows that ψ(a, c, x + y) < ψ(a, c, x) and ψ(a, c, x + y) < ψ(a, c, y). This means the inequality (2.7).

Similarly, the inequality (2.8) follows readily. The proof of Theorem 2.6 is complete.

Inequalities of the Turán type for the Mellin transform

Now we discover an inequality of the Turán type for the Mellin transform.

Theorem 3.1. For s > 0, the Mellin transform M (s) satisfies

F 2 (s + 1) ≤ F (s)F (s + 2). (3.1)
Proof. Applying the Hölder integral inequality finds that

M (αp + (1 -α)q) = ∞ 0 f (t)t αp+(1-α)q-1 d t = ∞ 0 f (t)t p-1 α f (t)t q-1 1-α d t ≤ ∞ 0 f (t)t p-1 d t α ∞ 0 f (t)t q-1 d t 1-α = M α (p)M 1-α (q).
This means that the Mellin transform M (s) is strictly logarithmically convex on (0, ∞). Further letting α = 1 2 , p = s, and q = s + 2 in the above inequality leads to the inequality (3.1). The proof of Theorem 3.1 is complete.

Example 3.1. Entry 17.43.26 in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] states that

M 1 (s) = M (cosech(x)) = 2(1 -2 -s )Γ(s)ζ(s), s > 1.
By Theorem 3.1, it follows readily that

M 2 1 (s + 1) ≤ M 1 (s)M 1 (s + 2
). After some simplification we acquire

ζ 2 (s + 1) ≤ s + 1 s (1 -2 -s )(1 -2 -s-2 ) (1 -2 -s-1 ) 2 ζ(s)ζ(s + 2), s > 1
which improves the Turán type inequality for the zeta function in [START_REF] Laforgia | Turán type inequalities for some special functions[END_REF].

Example 3.2. Entry 6.3.8 in [START_REF] Erdélyi | Tables of Integral Transforms[END_REF] states that When a = 1 in (3.2), we recover the Turán type inequality in [START_REF] Laforgia | Turán type inequalities for some special functions[END_REF].

Inequalities of the Turán type for the Laplace transform

Finally we find out an inequality of the Turán type for the Laplace transform. Proof. By the Hölder integral inequality, we have

L(αp + (1 -α)q) = ∞ 0 f (t)e -(αp+(1-α)q)t d t = ∞ 0 f (t)e -pt α f (t)e -qt 1-α d t ≤ ∞ 0 f (t)e -pt d t α ∞ 0 f (t)e -qt d t 1-α = L α (p)L 1-α (q).
In other words, the Laplace transform L(s) is strictly logarithmically convex on (0, ∞). Specially, setting α = 1 2 , p = s, and q = s + 2 in the above inequality leads to (4.1). The proof of Theorem 4.1 is complete.

Example 4.1. Entry 4.15.29 in [START_REF] Erdélyi | Tables of Integral Transforms[END_REF] states that

L 3 (s) = L (1 -e -t ) ν/2 J ν (a(1 -e -t ) 1/2 ) = Γ(s) 2 a s J ν+s (a)
for s > 0, a > 0, ν > -1, where J µ (z) denotes Bessel's functions. By Theorem 4.1, it follows that L 2 3 (s + 1) ≤ L 3 (s)L 3 (s + 2) which can be reformulated as

J 2 ν+s+1 (a) ≤ s + 1 s J ν+s (a)J ν+s+2 (a) (4.2)
for s > 0, 1 > a > 0, and ν > -1 2 . When taking ν = 0 and replacing s by s -1 for s ≥ 1 in (4.2), we derive an upper bound of the Turán type inequality in [5, Eq. (2.3)] for 0 < a < 1.

Example 4.2. Entry 4.3.11 in [START_REF] Erdélyi | Tables of Integral Transforms[END_REF] reads that

L 4 (s) = L (t 2 -a 2 ) ν-1/2 = 1 √ π Γ ν + 1 2 2a s ν K ν (as)
for s, a > 0 and ν > - for s, a > 0 and ν > -1 2 .

Remark 4.1. Many other Turán type inequalities can be obtained for functions whose Laplace and Mellin transforms exists. In particular, we can prove some Turán type inequalities for the gamma, beta, extended beta, hypergeometric, error, and compliment error functions.

Remark 4.2. This paper is a slightly revised version of the preprint [START_REF] Bhukya | Some inequalities of the Turán type for confluent hypergeometric functions of the second kind[END_REF].

M 2 (

 2 s) = M e -ax (1 -e -x ) -1 = Γ(s)ζ(s, a), s > 0, a > 0. By Theorem 3.1, we derive M 2 2 (s + 1) ≤ M 2 (s)M 2 (s + 2). After some simplification we acquire ζ 2 (s + 1, a) ≤ s + 1 s ζ(s, a)ζ(s + 2, a), s > 1, a > 0. (3.2)

Theorem 4 . 1 .

 41 The Laplace transform L(s) satisfies L 2 (s + 1) ≤ L(s)L(s + 2), s > 0.(4.1)

  1 2 , where K µ (z) denotes modified Bessel's functions. By Theorem 4.1, it follows that

	L 2 4 (s + 1) ≤ L 4 (s)L 4 (s + 2)
	which can be rewritten as
	K 2 ν (a(s + 1)) ≤	s 2 + 2s + 1 s(s + 2)

ν K ν (as)K ν (a(s + 2))

Acknowledgements

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


KOTTAKKARAN Gauhar Rahman

AND Sooppy Nisar
email: ksnisar1@gmail.com

Feng Qi

SOME NEW INEQUALITIES OF THE GR ÜSS TYPE FOR CONFORMABLE FRACTIONAL INTEGRALS

Keywords: 2010 Mathematics Subject Classification. Primary 26A33, Secondary 26D10, 26D15 Riemann-Liouville fractional integral, inequality of the Grüss type, conformable fractional integral, integral inequality, fractional integral operator

In the paper, the authors establish some new inequalities of the Grüss type for conformable fractional integrals. These inequalities generalize some known results.

Introduction

In [START_REF] Grüss | Über das Maximum des absoluten Betrages von 1 b-a b a f (x)g(x) d x -1 (b-a) 2 b a f (x) d x b a g(x) d x[END_REF], Grüss showed an integral inequality which connects the integral of the product of two functions and the product of integrals for these two function. This inequality reads that, if f and g are two continuous functions on [a, b] satisfying m ≤ f (τ ) ≤ M and n ≤ g(τ ) ≤ N for all τ ∈ [a, b] and m, M, n, N ∈ R, then

1 b -a b a f (τ )g(τ ) d t - 1 (b -a) 2 b a f (τ ) d t b a g(τ ) d t ≤ 1 4 (M -m)(N -n). (1)
For more information on the Grüss inequality [START_REF] Anastassiou | Opial type inequalities involving fractional derivatives of two functions and applications[END_REF], please refer to [11, Chapter X] and closely related references therein.

In the latest decades, the fractional integral inequalities involving the Riemann-Liouville fractional integrals have been widely studied by various researchers. The interested readers can refer to the work in [START_REF] Anastassiou | Opial type inequalities involving fractional derivatives of two functions and applications[END_REF][START_REF] Belarbi | On some new fractional integral inequalities[END_REF][START_REF] Dahmani | New inequalities in fractional integrals[END_REF][START_REF] Denton | Fractional integral inequalities and applications[END_REF][START_REF] Sarıkaya | On new inequalities via Riemann-Liouville fractional integrations[END_REF][START_REF] Sulaiman | Some new fractional integral inequalities[END_REF][START_REF] Zhu | Some new fractional q-integral Grüss-type inequalities and other inequalities[END_REF]. In [START_REF] Dahmani | New inequalities in fractional integrals[END_REF], Dahmani introduced the following fractional integral inequalities for the Riemann-Liouville fractional integrals: if f and g are two integrable functions on [0, ∞) satisfying m ≤ f (τ ) ≤ M and n ≤ g(τ ) ≤ N for all τ ∈ [0, ∞) and m, M, n, N ∈ R, then, for all τ, α, β > 0, we have

τ α Γ(α + 1) I α (f g)(τ ) -I α f (τ )I α g(τ ) ≤ τ α Γ(α + 1) 2 (M -m)(N -n) and τ α Γ(α + 1) I β (f g)(τ ) - τ β Γ(β + 1) I α (f g)(τ ) -I α f (τ )I β g(τ ) -I β f (τ )I α g(τ ) 2 ≤ M τ α Γ(α + 1) -I α f (τ ) I β f (τ ) -m τ β Γ(β + 1) + I α f (τ ) -m τ α Γ(α + 1) × M τ β Γ(β + 1) -I β f (τ ) N τ α Γ(α + 1) -I α g(τ ) I β g(τ ) -n τ β Γ(β + 1) + I α g(τ ) -n τ α Γ(α + 1) N τ β Γ(β + 1) -I β g(τ ) ,
where Γ is the classical Euler gamma function [START_REF] Nisar | Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function[END_REF][START_REF] Qi | Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function[END_REF][START_REF] Srivastava | Zeta and q-Zeta Functions and Associated Series and Integrals[END_REF] and the Riemann-Liouville fractional integral I µ of order µ ≥ 0 for a function f ∈ L 1 ((0, ∞), R) is defined [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF] by I 0 f (x) = f (x) and

I µ f (x) = 1 Γ(µ) x 0 (x -τ ) µ-1 f (τ ) d τ, (µ) > 0.
The Riemann-Liouville fractional integrals I µ α + and I µ β -of order µ > 0 are defined respectively by

I µ α + f (x) = 1 Γ(µ) x α (x -τ ) µ-1 f (τ ) d τ, x > α, (µ) > 0 (2)
and

I µ β -f (x) = 1 Γ(µ) β x (x -τ ) µ-1 f (τ ) d τ, x < β, (µ) > 0. ( 3
)
For more details about fractional integral operators ( 2) and ( 3), please refer to [START_REF] Dahmani | New generalization of Grüss inequality using Riemann-Liouville fractional integrals[END_REF][START_REF] Gorenflo | Fractional Calculus: Integral and Differential Equations of Fractional Order[END_REF][START_REF] Kilbas | Hadamard-type fractional calculus[END_REF][START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Qi | Some inequalities of Čebyšev type for conformable k-fractional integral operators[END_REF][START_REF] Samko | Fractional Integrals and Derivatives: Theory and Applications[END_REF][START_REF] Set | On new inequalities of Hermite-Hadamard-Fejér type for convex functions via fractional integrals[END_REF] and closely related references. The left and right sided fractional conformable integral operators are respectively defined [START_REF] Jarad | On a new class of fractional operators[END_REF] by

λ I µ a + f (x) = 1 Γ(λ) x a (x -a) µ -(τ -a) µ µ λ-1 f (τ ) (τ -a) 1-µ d τ (4)
and

λ I µ b -f (x) = 1 Γ(λ) b x (b -x) µ -(b -τ ) µ µ λ-1 f (τ ) (b -τ ) 1-µ d τ, (5)
where (λ) > 0. Obviously, if taking a = 0 and µ = 1, then (4) reduces to the Riemann-Liouville fractional integral [START_REF] Belarbi | On some new fractional integral inequalities[END_REF]. Similarly, if setting b = 0 and µ = 1, then (5) becomes the Riemann-Liouville fractional integral (3).

In [START_REF] Set | Chebyshev type inequalities involving new conformable fractional integral operators[END_REF] the conformable fractional integral

β I µ f (x) = 1 Γ(β) x 0 x µ -τ µ µ β-1 f (τ ) τ 1-µ d τ. (6)
was defined. From [START_REF] Gorenflo | Fractional Calculus: Integral and Differential Equations of Fractional Order[END_REF], one can obtain easily that α I µβ I µ f (x) = β+α I µ f (x) and

α I µβ I µ f (x) = β I µα I µ f (x).
In this paper, we will empoy the conformable fractional integral [START_REF] Gorenflo | Fractional Calculus: Integral and Differential Equations of Fractional Order[END_REF] to establish some new inequalities of the Grüss type for conformable fractional integrals.

Main results

We are now in a position to state and prove our main results.

Theorem 2.1. Let f be an integrable function on [0, ∞). Assume that there exist two integrable functions φ 1 , φ 2 on [0, ∞) such that

φ 1 (x) ≤ f (x) ≤ φ 2 (x), x ∈ [0, ∞). (7)
Then, for x, α, β > 0, we have

β I µ φ 1 (x) α I µ f (x)+ α I µ φ 2 (x) β I µ f (x) ≥ α I µ φ 2 (x) β I µ φ 1 (x)+ α I µ f (x) β I µ f (x). ( 8
)
Proof. From ( 7), for all τ, ρ ≥ 0, it follows that

[φ 2 (τ ) -f (τ )][f (ρ) -φ 1 (ρ)] ≥ 0.
Therefore, we have

φ 2 (τ )f (ρ) + φ 1 (ρ)f (τ ) ≥ φ 1 (ρ)φ 2 (τ ) + f (τ )f (ρ). (9)
Multiplying both sides of ( 9) by 1 Γ(α)

x µ -τ µ µ α-1 τ µ-1 and integrating over τ ∈ (0, x) lead to f (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 φ 2 (τ ) d τ + φ 1 (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 f (τ ) d τ ≥ φ 1 (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 φ 2 (τ ) d τ + f (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 f (τ ) d τ which gives f (ρ) α I µ φ 2 (x) + φ 1 (ρ) α I µ f (x) ≥ φ 1 (ρ) α I µ φ 2 (x) + f (ρ) α I µ f (x). ( 10
)
Multiplying both sides of ( 10) by 1 Γ(β)

x µ -ρ µ µ β-1 ρ µ-1 and integrating over ρ ∈ (0, x) result in α I µ φ 2 (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 f (ρ) d ρ + α I µ f (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 φ 1 (ρ) d ρ ≥ α I µ φ 2 (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 φ 1 (ρ) d ρ + α I µ f (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 f (ρ) d ρ.
which gives the required inequality [START_REF] Jarad | On a new class of fractional operators[END_REF].

From Theorem 2.1, we can derive the following two corollaries.

Corollary 2.1. Let f be an integrable function on [0, ∞) satisfying m ≤ f (x) ≤ M for all x ∈ [0, ∞) and m, M ∈ R. Then, for x, α, β > 0, we have

m x µβ µ β Γ(β + 1) α I µ f (x) + M x µα µ α Γ(α + 1) β I µ f (x) ≥ mM x µ(α+β) µ α+β Γ(α + 1)Γ(β + 1) + α I µ f (x) β I µ f (x). Corollary 2.2. Let f be an integrable function on [0, ∞) satisfying x µ ≤ f (x) ≤ x µ + 1 for all x ∈ [0, ∞).
Then, for x, α > 0, we have

2x µ(α+1) µ α Γ(α + 2) + x µα µ α Γ(α + 1) α I µ f (x) ≥ x µ(α+1) µ α Γ(α + 2) + x µα µ α Γ(α + 1) x µ(α+1) µ α Γ(α + 2) + [ α I µ f (x)] 2 .
Theorem 2.2. Let f and g be two integrable function on [0, ∞). Suppose that the inequality (7) holds and that there exist two integrable functions ψ 1 and

ψ 2 on [0, ∞) such that ψ 1 (x) ≤ g(x) ≤ ψ 2 (x), x ∈ [0, ∞). (11)
Then, for x, α, β > 0, the following four inequalities hold:

β I µ ψ 1 (x) α I µ f (x) + α I µ φ 2 (x) β I µ g(x) ≥ α I µ φ 2 (x) β I µ ψ 1 (x) + α I µ f (x) β I µ g(x), (12)
β I µ φ 1 (x) α I µ g(x) + α I µ ψ 2 (x) β I µ f (x) ≥ α I µ φ 1 (x) β I µ ψ 2 (x) + α I µ f (x) β I µ g(x), (13)
α I µ φ 2 (x) β I µ ψ 2 (x) + α I µ f (x) β I µ g(x) ≥ α I µ φ 2 (x) β I µ g(x) + β I µ ψ 2 (x) α I µ f (x), ( 14
) α I µ φ 1 (x) β I µ ψ 1 (x) + α I µ f (x) β I µ g(x) ≥ α I µ φ 1 (x) β I µ g(x) + α I µ f (x) β I µ ψ 1 (x). ( 15
)
Proof. From ( 7) and ( 11) and for x ∈ [0, ∞), we have

[φ 2 (τ ) -f (τ )][g(ρ) -ψ 1 (ρ)] ≥ 0.
Therefore, it follows that

φ 2 (τ )g(ρ) + ψ 1 (ρ)f (τ ) ≥ ψ 1 (ρ)φ 2 (τ ) + f (τ )g(ρ). ( 16
)
Multiplying both sides of ( 16) by 1 Γ(α)

x µ -τ µ µ α-1 τ µ-1 and integrating over τ ∈ (0, x) arrive at g(ρ)

x

0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 φ (τ ) d τ + ψ 1 (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 f (τ ) d τ ≥ ψ 1 (ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 φ 2 (τ ) d τ + g(ρ) x 0 1 Γ(α) x µ -τ µ µ α-1 τ µ-1 f (τ ) d τ which gives g(ρ) α I µ φ 2 (x) + ψ 1 (ρ) α I µ f (x) ≥ ψ 1 (ρ) α I µ φ 2 (x) + g(ρ) α I µ f (x). ( 17
)
Multiplying both sides of (17) by 1 Γ(β)

x µ -ρ µ µ β-1 ρ µ-1 and integrating over ρ ∈ (0, x) reveal

α I µ φ 2 (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 g(ρ) d ρ + α I µ f (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 ψ 1 (ρ) d ρ ≥ α I µ φ 2 (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 ψ 1 (ρ) d ρ + α I µ f (x) x 0 1 Γ(β) x µ -ρ µ µ β-1 ρ µ-1 g(ρ) d ρ.
which gives the required inequality [START_REF] Nisar | Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function[END_REF]. Making use of the inequalities

[ψ 2 (τ ) -g(τ )][f (ρ) -φ 1 (ρ)] ≥ 0, [φ 2 (τ ) -f (τ )][g(ρ) -ψ 2 (ρ)] ≤ 0, and [φ 1 (τ ) -f (τ )][g(ρ) -ψ 1 (ρ)
] ≤ 0, we can prove (13) to (15).

Corollary 2.3. Let f and g be two integrable functions on [0, ∞). Assume that there exist m, M, n, N ∈ R such that

m ≤ f (x) ≤ M, n ≤ g(x) ≤ N, x ∈ [0, ∞).
Then we have the following four inequalities

n x µβ µ β Γ(β + 1) α I µ f (x) + M x µα µ α Γ(α + 1) β I µ g(x)
≥ nM x µ(α+β) µ α+β Γ(α + 1)Γ(β + 1)

+ α I µ f (x) β I µ g(x), m x µβ µ β Γ(β + 1) α I µ g(x) + N x µα µ α Γ(α + 1) β I µ f (x) ≥ mN x µ(α+β) µ α+β Γ(α + 1)Γ(β + 1) + β I µ f (x) α I µ g(x), M N x µ(α+β) µ α+β Γ(α + 1)Γ(β + 1) + β I µ f (x) α I µ g(x) ≥ M x µα µ α Γ(α + 1) β I µ g(x) + N x µβ µ β Γ(β + 1) α I µ f (x),
and

mn x µ(α+β) µ α+β Γ(α + 1)Γ(β + 1) + β I µ f (x) α I µ g(x) ≥ m x µα µ α Γ(α + 1) β I µ g(x) + n x µβ µ β Γ(β + 1) α I µ f (x).
Theorem 2.3. Let f be an integrable function on [0, ∞) and let φ 1 and φ 2 be two integrable functions on [0, ∞). Assume that the inequality (7) holds. Then, for x, α > 0, we have

x µα µ α Γ(α + 1) α I µ f 2 (x) -[ α I µ f (x)] 2 = [ α I µ φ 2 (x) -α I µ f (x)][ α I µ f (x) -α I µ φ 1 (x)] - x µα µ α Γ(α + 1) α I µ [φ 2 (x) -f (x)][f (x) -φ 1 (x)] + x µα µ α Γ(α + 1) α I µ φ 1 f (x) -α I µ φ 1 (x) α I µ f (x) + x µα µ α Γ(α + 1) α I µ φ 2 f (x) -α I µ φ 2 (x) α I µ f (x) + α I µ φ 1 (x) α I µ φ 2 (x) - x µα µ α Γ(α + 1) α I µ φ 1 φ 2 (x). ( 18
)
Proof. For τ, ρ > 0, we have

[φ 2 (ρ) -f (ρ)][f (τ ) -φ 1 (τ )] + [φ 2 (τ ) -f (τ )][f (ρ) -φ 1 (ρ)] -[φ 2 (τ ) -f (τ )][f (τ ) -φ 1 (τ )] -[φ 2 (ρ) -f (ρ)][f (ρ) -φ 1 (ρ)] = f 2 (τ ) + f 2 (ρ) -2f (τ )f (ρ) + φ 2 (ρ)f (τ ) + φ 1 (τ )f (ρ) -φ 1 (τ )φ 2 (ρ) + φ 2 (τ )f (ρ) + φ 1 (ρ)f (τ ) -φ 1 (ρ)φ 2 (τ ) -φ 2 (τ )f (τ ) + φ 1 (τ )φ 2 (τ ) -φ 1 (τ )f (τ ) -φ 2 (ρ)f (ρ) + φ 1 (ρ)φ 2 (ρ) -φ 1 (ρ)f (ρ). ( 19
)
Multiplying both sides of ( 19) by 1 Γ(α)

x µ -τ µ µ α-1 τ µ-1 and integrating over τ ∈ (0, x) yield [φ 2 (ρ) -f (ρ)][ α I µ f (x) -α I µ φ 1 (x)] + [ α I µ φ 2 (x) -α I µ f (x)][f (ρ) -φ 1 (ρ)] -α I µ [φ 2 (x) -f (x)][f (x) -φ 1 (x)] -[φ 2 (ρ) -f (ρ)][f (ρ) -φ 1 (ρ)] x µα µ α Γ(α + 1) = α I µ f 2 (x) + f 2 (ρ) x µα µ α Γ(α + 1) -2f (ρ) α I µ f (x) + φ 2 (ρ) α I µ f (x) + f (ρ) α I µ φ 1 (x) -φ 2 (ρ) α I µ φ 1 (x) + f (ρ) α I µ φ 2 (x) + φ 1 (ρ) α I µ f (x) -φ 1 (ρ) α I µ φ 2 (x) -α I µ φ 2 f (x) + α I µ φ 1 φ 2 (x) -α I µ φ 1 f (x) -φ 2 (ρ)f (ρ) x µα µ α Γ(α + 1) + φ 1 (ρ)φ 2 (ρ) x µα µ α Γ(α + 1) -φ 1 (ρ)f (ρ) x µα µ α Γ(α + 1)
.

Multiplying both sides of (20) by 1 Γ(α)

x µ -ρ µ µ α-1 ρ µ-1 and integrating the resultant identity with respect to ρ from 0 to x bring out

[ α I µ φ 2 (x) -α I µ f (x)][ α I µ f (x) -α I µ φ 1 (x)] + [ α I µ φ 2 (x) -α I µ f (x)] ×[ α I µ f (x) -α I µ φ 1 (x)] -α I µ [φ 2 (x) -f (x)][f (x) -φ 1 (x)]
x µα µ α Γ(α + 1)

-α I µ [φ 2 (ρ) -f (ρ)][f (ρ) -φ 1 (ρ)] x µα µ α Γ(α + 1) = x µα µ α Γ(α + 1) α I µ f 2 (x) + x µα µ α Γ(α + 1) α I µ f 2 (x) -2 α I µ f (x) α I µ f (x) + α I µ φ 2 (x) α I µ f (x) + α I µ f (x) α I µ φ 1 (x) -α I µ φ 2 (x) α I µ φ 1 (x) + α I µ f (x) α I µ φ 2 (x) + α I µ φ 1 (x) α I µ f (x) -α I µ φ 1 (x) α I µ φ 2 (x) - x µα µ α Γ(α + 1) α I µ φ 2 f (x) + x µα µ α Γ(α + 1) α I µ φ 1 φ 2 (x) - x µα µ α Γ(α + 1) α I µ φ 1 f (x) - x µα µ α Γ(α + 1) α I µ φ 2 f (x) + x µα µ α Γ(α + 1) α I µ φ 1 φ 2 (x) - x µα µ α Γ(α + 1) α I µ φ 1 f (x)
which yields the required inequality [START_REF] Set | On new inequalities of Hermite-Hadamard-Fejér type for convex functions via fractional integrals[END_REF].

Corollary 2.4. Let f be an integrable function on [0, ∞) satisfying m ≤ f (x) ≤ M for all x ∈ [0, ∞). Then, for all x, α > 0, we have

x µα µ α Γ(α + 1) α I µ f 2 (x) -[ α I µ f (x)] 2 = M x µα µ α Γ(α + 1) -α I µ f (x) × α I µ f (x) -m x µα µ α Γ(α + 1) - x µα µ α Γ(α + 1) α I µ [M -f (x)][f (x) -m].
Theorem 2.4. Let f, g, φ 1 , φ 2 , ψ 1 , and ψ 2 be integrable functions on [0, ∞) satisfying [START_REF] Grüss | Über das Maximum des absoluten Betrages von 1 b-a b a f (x)g(x) d x -1 (b-a) 2 b a f (x) d x b a g(x) d x[END_REF] and (11) on [0, ∞). Then, for all x, α > 0, we have

x µα µ α Γ(α + 1) α I µ f g(x) -α I µ f (x) α I µ g(x) ≤ T (f, φ 1 , φ 2 )T (g, ψ 1 , ψ 2 ) , (21)
where

T (u, v, w) = [ α I µ w(x) -α I µ u(x)][ α I µ u(x) -α I µ v(x)] + x µα µ α Γ(α + 1) α I µ vu(x) -α I µ v(x) α I µ u(x) + x µα µ α Γ(α + 1) α I µ wu(x) -α I µ w(x) α I µ u(x) + α I µ v(x) α I µ w(x) - x µα µ α Γ(α + 1) α I µ vw(x). Proof. Define H(τ, ρ) = [f (τ ) -f (ρ)][g(τ ) -g(ρ)], τ, ρ ∈ (0, x), x > 0. ( 22
)
Multiplying both sides of ( 22) by 1 Γ 2 (α)

x µ -τ µ µ α-1 x µ -ρ µ µ α-1 τ µ-1 ρ µ-1 for τ, ρ ∈ (0, x) and integrating the desired inequality with respect to τ, ρ from 0 to x yield

1 2Γ 2 (α) t 0 t 0 x µ -τ µ µ α-1 x µ -ρ µ µ α-1 τ µ-1 ρ µ-1 H(τ, ρ) d τ d ρ = x µα µ α Γ(α + 1) α I µ f g(x) + α I µ f (x) α I µ g(x). (23)
Applying the Cauchy-Schwartz inequality to (23) leads to

x µα µ α Γ(α + 1) α I µ f g(x) + α I µ f (x) α I µ g(x) 2 ≤ x µα µ α Γ(α + 1) α I µ f 2 (x) -[ α I µ f (x)] 2 x µα µ α Γ(α + 1) α I µ g 2 (x) -[ α I µ g(x)] 2 . ( 24
) Since [φ 2 (x) -f (x)][f (x) -φ 1 (x)] ≥ 0 and [ψ 2 (x) -g(x)][g(x) -ψ 1 (x)] ≥ 0 for x ∈ [0, ∞), we have x µα µ α Γ(α + 1) [φ 2 (x) -f (x)][f (x) -φ 1 (x)] ≥ 0 and x µα µ α Γ(α + 1) [ψ 2 (x) -g(x)][g(x) -ψ 1 (x)] ≥ 0.
Thus, from Theorem 2.3, we obtain

x µα µ α Γ(α + 1) α I µ f 2 (x) -[ α I µ f (x)] 2 ≤ [ α I µ φ 2 (x) -α I µ f (x)][ α I µ f (x) -α I µ φ 1 (x)] +
x µα µ α Γ(α + 1)

α I µ φ 1 f (x) -α I µ φ 1 (x) α I µ f (x) +

x µα µ α Γ(α + 1) α I µ φ 2 f (x) -α I µ φ 2 (x) α I µ f (x) + α I µ φ 1 (x) α I µ φ 2 (x) -x µα µ α Γ(α + 1)

α I µ φ 1 φ 2 (x) = T (f, φ 1 , φ 2 ). ( 25)

Similarly, we have

x µα µ α Γ(α + 1) α I µ g 2 (x) -[ α I µ g(x)] 2 ≤ [ α I µ ψ 2 (x) -α I µ g(x)][ α I µ g(x) -α I µ ψ 1 (x)] + x µα µ α Γ(α + 1) α I µ ψ 1 g(x) -α I µ ψ 1 (x) α I µ g(x) + x µα µ α Γ(α + 1) α I µ ψ 2 g(x)

α I µ ψ 2 (x) α I µ g(x)+ α I µ ψ 1 (x) α I µ ψ 2 (x)-x µα µ α Γ(α + 1)

α I µ ψ 1 ψ 2 (x) = T (g, ψ 1 , ψ 2 ).

(26)

Combining ( 24), (25), and (26), we obtain the desired inequality (21).

Remark 2.1. For m, M, n, N ∈ R, if T (f, φ 1 , φ 2 ) = T (f, m, M ) and T (g, ψ 1 , ψ 2 ) = T (g, n, N ), then the inequality (21) reduces to

x µα µ α Γ(α + 1)

α I µ f g(x) -α I µ f (x) α I µ g(x) ≤ x µα 2µ α Γ(α + 1)

2 (M -m)(N -n).
in [START_REF] Set | Grüss type inequalities involving new conformable fractional integral operators[END_REF]Theorem 1].

Remark 2.2. In this paper, we presented some new conformable fractional integral inequalities which generalize those corresponding ones in [START_REF] Tariboon | Some new Riemann-Liouville fractional integral inequalities[END_REF].

Remark 2.3. This paper is a slightly revised version of the preprint [START_REF] Rahman | Some new inequalities of the Grüss type for conformable fractional integrals[END_REF].


P W Hoffrogge

L A Barrales-Mora

Grain reconstruction and determination of grain attributes from MD data sets of FCC polycrystals

Keywords: molecular-dynamics, grain reconstruction, post-processing, parallelization, grain growth, grain rotation

An algorithm for the identification and reconstruction of crystallites from molecular dynamics data sets of FCC materials was developed. The algorithm was tested in an Al polycrystal possessing a weak texture. For the conditions tested, the algorithm was able to find all of the input orientations and reconstruct the grains according to their crystallographic orientation. The method is capable of calculating specific attributes of grains, namely, volume, center of mass, average orientation and orientation spread. The method additionally provides a mapping method to track grains during time-row dataset. The software uses shared-memory parallelization implemented by the use of the OpenMP-API, which enables a fast analysis of large volumes of data with excellent performance.

Introduction

In molecular dynamics (MD) simulations, the analysis and interpretation of the data remain a substantial and difficult part. Certain tools are required in order to reduce the amount of information and obtain useful quantities. In several situations, information on the evolution of 3D features is necessary, for example, grains in polycrystalline aggregates. This information is, however, not delivered by most of the available methods for the characterization of MD data sets. This task remains difficult because it involves identifying crystallographically the environment of the atoms to assign an orientation respective to a coordinate system and to reconstruct from this information the volume with identical orientation i.e., the grains. Due to the mathematical complexity of rotations and reconstruction algorithms, these calculations are slow and costly for their execution during runtime and even during post-processing if large data sets are involved. Nevertheless, several methods which enable the identification of crystal structures from atomic-based data have been already developed. The most straightforward way to distinguish between some of these is to compute the coordination number, i.e. the number of nearest neighbors (NNs) or next-nearest neighbors (NNNs) of the atoms. While this is a fast tool and also easy to implement, it obviously cannot distinguish between different atomic structures exhibiting the same coordination number. Another technique which allows a much more reliable identification of crystalline structures was first introduced by Honeycutt and Andersen [1]. This method analyzes the extent and connectivity properties of atom-diagrams which are comprised of nearest neighbors to two adjacent atoms. Based on this principle, also known as Common Neighbor Analysis or CNA, several algorithms have been implemented [2,3,4] and successfully performed to MD simulation data. Similarly, the centro-symmetry parameter, energy filters , bond order and angle and Voronoi analysis can also be used to discriminate atoms arranged in specific crystal structures [2]. Nevertheless, none of these methods can distinguish between groups of atoms with the same crystallographic orientation. To differentiate between differently oriented crystals, the definition of order parameters has been utilized. The drawbacks of this approach are that the orientations ought to be known beforehand and that complex rotations of the crystals might out-range the scope of the parameter. Nevertheless, this approach has been successfully utilized in numerous simulations to track, for instance, grain boundaries [5,6,7,8,9] and grain rotation [10,11,12,13,14].

The motivation of the present contribution is to introduce a method that allows the determination of the relative orientation of groups of atoms and reconstruct the grain from only this information. The method is able to generate a space-resolved three dimensional grain decomposition of atom-position data sets for FCC materials, such as generated by MD-simulations. The first step of the method calculates an orientation for each atom by taking into account the positions of the nearest-neighbor atoms. In a second step, the method generates grain entities from atoms with similar orientations by collecting them via nearest-neighbor paths. The method is capable of calculating grain specific properties, namely volume, center of mass, average orientation and orientation spread. The method additionally provides a mapping method in order to track grains during a time-row data set. The method is introduced in order to be able to track the grain growth evolution of individual grains during MD-simulations, additionally enabling a grain-resolved visualization of polycrystalline FCC data sets.

Methods

To identify the grains from a data set containing atomic positions, it is necessary to determine first the orientation of the atoms as a per-atom attribute from the local neighborhood. Once an atom (referred to as central atom) is selected arbitrarily, the calculation of the orientation proceeds in the following way:

(i) Identify all the nearest neighbors and calculate the relative positions of the neighbors to the corresponding central atom.

(ii) Determine the affine 3×3 transformation matrix M from the reference (non-rotated case) to the current crystal-configuration.

(iii) Determine the rotation that best fits M as a least square solution.

All of these steps have an inherent mathematical and/or programmatic complexity. In the present section, we will discuss the algorithms that were utilized to solve each of these problems. In the appendix, these solutions are presented in the form of pseudocode whereas the program is offered as an open-source project [15].

Orientation determination

The calculation begins with the identification of the nearest-neighbors. Once the neighbors of an atom have been identified, displacement vectors are calculated with the current atom as origin. These vectors are then normalized so that only unit vectors are used for the orientation calculation. Since in the FCC crystal structure each neighbor has an antipodal counterpart, the list of nearest neighbor vectors is reduced by finding vectors with a near 180 • -relationship. The resulting mean direction v r is calculated by vector subtraction of any v and its antipodal partner v a :

v r = 1 2 ( v -v a ) (1)
In the FCC lattice, there are three pairs of neighbor vectors, each lying on different {100} planes. Since each vector ought to have only one defined perpendicular partner in the FCC-case, we try to find the directions of this planes by identifying vector pairs with a near 90 • -relationship. Now, at least six different vectors must be available in order to find all three pairs. Afterwards, the 100 directions v 100 are calculated as the cross product of both pairs vectors v 1 and v 2 :

v 100 = v 1 × v 2 (2)
The transformation matrix M can now easily be obtained as:

M =    v x 100 v y 100 v z 100 v x 010 v y 010 v z 010 v x 001 v y 001 v z 001    (3)
The assignment of the vectors to a certain row is of arbitrary nature, which can cause a negative determinant. It is possible to avoid this issue by calculating the determinant and checking whether it is negative or not. In the case that it results negative, the method inserts -v 100 in the first row instead of v 100 . The matrix M is not necessarily a pure rotation matrix because the atoms can assume non perfect lattice positions due to elastic distortions or thermal vibrations. For this reason, we have to estimate the rotation that best fits the affine transformation defined by our matrix M . This problem is well known in aeronautics and thus, several solutions have been already provided in the past. For instance, Horn [16] introduced a direct method that estimates an orientation from a set of corresponding vectors in two coordinate systems for aeronautical purposes.

A similar method that requires computing the eigenvalues of a 4×4 matrix was presented by Bar-Itzhack [17]. This method is based on the q-method proposed by Keat [START_REF] Keat | Analysis of least-squares attitude determination routine doaop[END_REF]. We opted to use this method in the present contribution as operations with quaternions are computationally less expensive. The q-method utilizes a cost function c to be minimized:

c = 1 2 k i=1 a i | b i -D(q) r i | 2 (4)
where r i denotes a unit vector in the reference/sample coordinate system and b i the same vector in the body/crystal coordinate system. In turn, q is the sought rotation quaternion, which defines the rotation matrix D whereas a i is an optional weight to each of the vector pairs. The cost function contains the square lengths of the error vectors, i.e. the minimization is applied with the Euclidean norm. In [START_REF] Keat | Analysis of least-squares attitude determination routine doaop[END_REF], it was shown that the solution of this problem is reduced to the computation of the eigenvalues of a symmetrical 4 × 4 matrix, which is defined by the set of vectors r i and b i . To find the best-fitting rotation quaternion to the transformation matrix, the method introduced in [17] was utilized. In Bar-Itzhack's contribution, it was shown that by utilizing the three base vectors [100], [010] and [001] of the reference coordinate system, the problem can be formulated for non-orthogonal matrices. As a result, the new matrix M 4 can be computed from the values of the matrix M from Eq. (3) as:

M 4 = 1 3      r 11 -
     (5)
The quaternion q representing the rotation that best fits M is the eigenvector of the most positive eigenvalue of M 4 . The quaternion q is assigned to the current atom during the calculation. In the case that the matrix M cannot be defined owing to strong lattice distortions, the corresponding atom is marked as non-oriented.

Grain detection

After the assignation of the orientation to the atoms, the grains are identified in a second step. The identification method uses the orientation information generated in the previous step. The procedure begins from an initial atom as a seed and recursive assignations are applied to nearest-neighbor atoms. For this, the following steps are conducted:

(i) Check whether the nearest-neighbor atoms to a seed atom:

(a)
are not yet assigned to another grain-entity.

(b) own an orientation close to the seed-atom according to a user defined threshold (1 • as default). This is referred to as the local criterion. (c) own an orientation close (< 3 • as default) to the mean orientation of the current grain if and only if enough atoms have been aggregated to this grain. This is referred to as the global criterion.

(ii) If these conditions are fulfilled, the neighbor atom is aggregated. The procedure is now repeated from the beginning with the aggregated atoms as seeds.

This procedure is exercised for every atom in the data set. It can occur during an iteration that the number of aggregated atoms of a grain is smaller than a reasonable amount (200 atoms as default). Whenever this happens, the corresponding grain information is removed and all collected atoms during this step are set to be unassigned again.

(a) (b) (c) (d) In Fig. 2, the application of the misorientation criteria is exemplified. The local criterion compares the orientation of two atoms in a nearest-neighbor relationship whereas the global criterion assures that the orientation of an atom to be aggregated has a similar orientation than that of the current grain. The global criterion allows for the detection of grains separated by low angle grain boundaries, which otherwise cannot be detected only by local comparisons due to the gradual change of orientation between grains at positions far away from dislocation centers.

Orphan atom adoption

In some cases owing to strong thermal vibrations or proximity to lattice defects, the determination of the orientation can fail or deliver orientations different than those found in an immediate neighborhood. In both cases, such atoms will not be assigned to a grain entity during the grain identification step. These atoms, deemed in the following igure 2: A fully identified grain (a) consists of the atoms (solid circles) enclosed by the dashed line after few iterations. Nearest-neighbors to the already aggregated atoms are tested in subsequent iterations. In order to collect an atom to the current grain entity, the atom must have an orientation difference less than 1 • with respect to the seed atom (b) and additionally ensure that the orientation difference with respect to the mean orientation of the grain is less than 3 • (c).

as orphan atoms, can cause a substantial underestimation of the volume of the grains. To solve this problem, we utilized a method inspired in noise correction methods used in EBSD mapping algorithms [START_REF] Humphreys | [END_REF]20]. The approach consists of the following steps:

(a) For each orphan atom, the grains to which the nearest neighbors belong are identified and counted.

(b) From the list of grains determined in the previous step, the one with most occurrences is selected.

(c) If the maximum number of occurrences is higher than a user defined value (the default is three), the orphan atom is assigned to the grain selected in step (b).

(d) The procedure is repeated for all remaining orphan atoms.

The minimum number of grain occurrences (three) was introduced in order to suppress random grain assignations. This procedure is schematically shown in Fig. 3. Note that initially (Fig. 3b) only orphan atoms a A and a B are adopted because only they had enough nearest neighbors assigned to a common grain. This procedure is applied iteratively so that the number of orphan atoms is reduced to a minimum.

Grain-tracking over time

In order to determine the evolution of grain properties over time, it is necessary to track the grains from snapshots generated at different times. To do this, we assume that geometrical and orientation attributes of the grains show little change between subsequent time states. This enables finding correspondence of grains by comparing calculated properties. We utilized two criteria to identify possible corresponding candidates, namely the center-of-mass (COM) and the misorientation between grain candidates. To begin with, the COM of the grains is calculated for the data sets generated at times t and t + ∆t. An arbitrary grain G 1 is selected from the second data set (t + ∆t). From the first one (t), the grains that fulfill d ≤ d max , where d is the distance to grain G 1 and d max is a user defined threshold value, are chosen. This list is further refined by rejecting grains with a misorientation larger than a user-defined value (∆g > 5 • as default). If more than one grain exist that fulfill both criteria, the grain with the shortest distance d is chosen. Initially, many atoms close to grain boundaries are not assigned to a specific grain. These atoms are adopted recursively depending on the frequency occurrences to neighboring grains (b). For example, in (c) atom a A is adopted by the grain on the left whereas atom a B is adopted by the grain on the right. After a few iterations the number of orphan atoms is reduced to a minimum (d).

(a) (b) (c) (d) a A a B a A a B
The methods described in the previous sections are provided as pseudo-code in the appendix and implemented as a command-line program called GraDe-A (Grain Detection Algorithm). The program was written in the object-oriented C++ programming language. Furthermore, the software uses shared-memory parallelization by means of the OpenMP-API [21], which enables analyzing several input files in parallel. The code has been written accounting for computation time considerations. Thereby, the utilization of computationally slow operations such as square root or cosine calculations has been avoided. Additionally, the computation domain has been split into cells in order to accelerate nearest-neighbor-search algorithms. All orientation data is internally stored as unit quaternions with memory consumption for only four floatingpoint numbers. Therefore, all misorientation calculations are implemented as simple quaternion operations, which are superior to rotation matrix operations with regard to computational effort. Unique cubic orientation quaternions [22] are used in order to be able to calculate the mean-orientation and the orientation spread [22] for each grain. Additionally, this enables to use a fast disorientation calculation for the grain identification algorithm. The code is also offered as open-source software [15].

Molecular dynamics simulations of grain growth

Molecular dynamics simulations were employed to test the developed software. For this purpose, nanocrystalline grain growth was simulated. The large-scale atomic/molecular massively parallel simulator (LAMMPS) code [23] was utilized. The atomic interactions were described by the EAM potential for Al developed by Mishin et al. [11]. The algorithms presented in previous sections were used to determine the orientation of grains in an Al polycrystal and reconstruct them in time-resolved molecular dynamics data sets [15]. The simulation block used in these MD simulations was composed of one hundred Voronoi grains with different orientations (Fig. 4) that were defined so that a weak texture was predominant in the microstructure. The orientations were assigned randomly to the grains. The simulation box had a side length of 43.27 nm in all directions and contained 4834137 atoms for an initial average grain size of approximately 11.57 nm. Periodic boundary conditions were used on all the surfaces of the simulation box. Before the MD simulation was performed, the energy of the system was minimized by the conjugate-gradient algorithm. Subsequently, damped dynamics were applied to fully relax the grain boundaries. The isothermal-isobaric (NPT) ensemble was used for the time integration with a time step of 0.2 fs. The simulation was performed at 500 • C. The total simulation time corresponded to 4 ns. In order to warrant the accuracy of the measured properties of the simulated systems, snapshots were regularly saved in intervals of 0.01 ns. The obtained configurations were subsequently quenched computationally in an NPT-ensemble for a time of 0.1 ns to a target temperature of 1 K and additionally relaxed by the conjugate gradient algorithm after setting the kinetic energy to zero. For the identification of the grains, the default thresholds were consistently used for all of the data analyzed. An amount of 200 atoms was set as a minimum for any grain detection.

The MD-simulations were performed on the Jülich Supercomputing Cluster and the post-processing of the data on the RWTH-Aachen Computer Cluster.

Results

The algorithm was benchmarked on a Bull MPI-S machine possessing Intel Xeon X5675 processors. For these tests, we used 24 files of approximately 240 MB containing ∼4.8 million atoms. The utilized parallelization scheme allowed reducing the computation time from 1.02 hours for two threads to only 12 minutes for 12 threads (Table 1). Irregular scaling with the number of threads was observed. This behavior was caused by an asymmetrical distribution of tasks per thread seen in situations where the number of tasks is not a multiple of the number of threads e.g. 10 threads. This problem is less significant with an increasing number of tasks. For instance, in the present contribution a total of 400 files for a total data volume of 140 GB were evaluated in only approximately 4 hours. To estimate the accuracy of the algorithm, the determined orientations were compared to the ones used as input. It is stressed that the algorithm does not require initial orientations as input. The figures plotted in Fig. 4 illustrate clearly that the algorithm is capable to identify correctly the orientations of the grains. Fig. 4a and Fig. 4b correspond respectively to the seeded orientations and the orientations calculated from the MD data set. Fig. 4c shows the microstructure colored after the identification number of the grains, which is a number determined by the sorted volume of the grains. This is the reason, why large grains appear blue and small red. The identification number can change from one snapshot/time-step to another due simply to the growth and shrinkage of the grains. However, as explained in the previous section the algorithm is capable of tracking correctly the grains as the COM and orientation are the criteria utilized for the tracking, independently of the identification number assigned at a time step.

Clearly, the advantage of the grain reconstruction is that it makes possible the tracking of grain properties such as volume, center of mass, average orientation, orientation spread and orientation change. Additionally, the algorithm permits the identification of sudden events that influence decisively the development of the microstructure. For instance, Fig. 5a illustrates the global kinetics obtained by detecting the remaining grains at each snapshot of the simulation. It is noted that since grains are individually identified, other algorithms for grain size measurements, such as fitting ellipsoids to the grains [24], can be easily implemented. c), where the grains are colored according to an integer number that serves as an id for the grains. This number was assigned so that the smallest number corresponds to grains with the largest volume and vice versa.

In turn, Fig. 5b shows the individual kinetics of the detected grains. Note that the ordinate is in logarithm scale so that the collapse of the grains is better visualized. For comparison, the orientations of the remaining grains after the finalization of the simulation are plotted in the pole figures shown in Fig. 5c; a slight sharpening of the initially used weak texture is evident as expected for the texture evolution after grain growth [START_REF] Barrales-Mora | 2D and 3D Grain Growth Modeling and Simulation[END_REF]. Since the algorithm is capable to determine a per-atom orientation and identify the grains, it is also possible to determine any change in orientation that might be induced during grain growth. This is a topic of interest because Cahn and Taylor [START_REF] Cahn | [END_REF] predicted that rotation of grains can be caused by the motion of grain boundaries. Several simulation studies [12,13,27,14,28] and some experimental ones [29,30] have confirmed this effect for stressed nanocrystalline polycrystals. However, other experimental studies [31] have not observed this concomitant grain migration and rotation in systems closer to the simulated ones, where rotation is thought to be more prominent. In fact, in previous simulation studies [14,32], we showed that rotation is strongly dependent on the character of the grain boundary. Furthermore, most of the computational studies have been performed in bi-or tricrystals [27] and with CSL grain boundaries as a necessity to fulfill periodicity on all surfaces of the simulation setups. However, grain boundaries in real microstructures tend to be of a mixed and general character and deviate substantially from CSL or Σ grain boundaries. For this reason it is unclear whether rotation will be observed as a generalized mechanisms of microstructure evolution. To investigate if grain rotation is observed in nanocrystalline polycrystals subjected to grain growth, we tracked the orientation change of the grains. In Fig. 6a the average absolute change of grain orientation as a function of time is shown. The maximum orientation change was less than 3.5 • ; this evinced only a marginal orientation change that is not in accordance with Cahn and Taylor theoretical expectations [START_REF] Cahn | [END_REF]14,32,13]. The reason for this may be related to the structure of the grain boundaries as discussed in [14] but a more detailed investigation of the mechanisms of grain boundary migration is necessary to affirm this. As for the kinetics, the change of orientation can also be resolved for individual grains. This is illustrated in Fig. 6b, where the rotation of all the grains in the simulations is plotted as a function of time. Interestingly, rotation is most prominent for grains that are close to collapse as evinced by those grains, whose orientation change increases in a very short time.

To give an impression of the practicality of the algorithm, individual grain kinetics and orientation change for an arbitrary grain are shown in Fig. 6c and Fig. 6d. This representation makes possible identifying significant but rare events that are determinant for microstructure evolution [33,34,35,36]. In the case of this grain, the kinetics showed an initial fast growth until it reached a constant volume from approximately 2 to 2.5 ns. Afterwards, the grain shrank until the simulation finished. The morphology of this grain can be seen in Fig. 7. This grain was able to survive the whole simulation and grew successfully for most of its life reaching a maximum grain size after 2.66 ns (Fig. 7c). This growth is also substantiated by the number of faces that the grain gained. For instance, after 1.33 ns (Fig. 7b) the grain had significantly more faces than originally (Fig. 7a) and evidently a concave shape that signifies positive growth rates. At the last simulated time (Fig. 7d), the grain had already started shrinking showing a smaller size than at a previous time (Fig. 7c) and also a convex shape. Note that the shape is not irregular and approximates excellently grains observed in polycrystals by serial-sectioning or tomography [37,38,39]. This is also an indication that the software can identify and reconstruct the grains adequately. In the same manner, the evolution of single grains showing interesting features can be followed visually by discriminating them from the data set of identified grains. For instance, this tool was already utilized to investigate size-effects observed in nanocrystalline materials; this will be the issue of a future contribution.

The purpose of this manuscript was to introduce the computational tool GraDe-A for the determination, reconstruction and tracking of grains in FCC polycrystals from MD data sets. This tool can be utilized to evaluate the change of properties of the grains and we expect it to become a valuable tool for the interpretation of data from MD simulations. Recently, also a tool capable of discerning orientation from MD datasets was developed [40], which in conjunction with the tool here introduced would allow analyzing grains in polycrystals with different crystal structures.

Summary

A computational tool, which is capable of identifying grain entities by the calculation of per-atom orientation quaternions, was developed. It was demonstrated that the tool is powerful enough to allow for grain-resolved 3D-visualizations of the grains and that it provides useful grain-quantities, such as mean-orientation, volume and rotation angles. A Voronoi-tesselated microstructure with a weak texture was synthesized and used to simulate grain growth at 500 • C by molecular dynamics; snapshots from these simulations were used to benchmark the computational tool. The developed algorithm was used to determine global and per-grain kinetics and mean rotation of the grains. Additionally to these capabilities, there are various improvements and extensions that are currently under development:

• Add support for other lattice-structures

• Identifying and highlighting other network-entities such as faces, edges and junctions

• Calculate grain-properties such as triple line length, turning angle and mean width

• Obtaining topological and connectivity-information for each grain

• Calculating the 5 parameter space of grain boundaries

• Determination of Bernal structures of grain boundaries

• Automatically identify grain-coalescence events These enhancements would be helpful for the analysis of nanocrystalline grain growth and rotation since most of the processes observed are difficult to discuss without knowing exact information on the involved defects and geometries.

1/2(v i -v j ) of near antipodal pairs. If a sufficient amount of directions remains, the orientation calculator utilizes them to formulate a transformation matrix M , as described in section 2.1. In order to obtain the largest eigenvalue as the resulting quaternion q we use the external open-source linear-algebra solver Armadillo §. § http://arma.sourceforge.net Algorithm A1: Algorithm calculating the orientation of an atom in a fcc-lattice structured material by taking into account the near neighborhood.

Input: The antipodal threshold angle ∆α and the perpendicular threshold angle ∆β in the current work chosen to be 5 • and 11.5 • , respectively. Use algorithm A2 to generate the reduced vector list V R containing n R vectors // For the fcc-lattice at least 6 neighbor directions need to be available

Figure 1 :

 1 Figure 1: Sequence for the identification of the grains. (a) A seed atom is selected from the dataset; open circles represent atoms with the same orientation as the seed atom. (b) The first neighbors to the seed are identified by comparison of the per-atom orientations and collected into a grain entity. (c-d) In subsequent iterations, the already identified atoms (green in (c), brown in (d)) act as seeds for their next-neighbors. Atoms are continually collected until no equally oriented atoms can be found in an iteration step.

Figure 3 :

 3 Figure 3: Adoption of orphan atoms. The orphaned state is represented by open circles whereas differently colored circles signify atoms aggregated to different grains. (a)Initially, many atoms close to grain boundaries are not assigned to a specific grain. These atoms are adopted recursively depending on the frequency occurrences to neighboring grains (b). For example, in (c) atom a A is adopted by the grain on the left whereas atom a B is adopted by the grain on the right. After a few iterations the number of orphan atoms is reduced to a minimum (d).

Figure 4 :

 4 Figure 4: (a) The input orientation are shown in the pole figures and (b) comparedto the determined orientations by the GraDe-A tool. An excellent agreement can be observed. The reconstruction of the grains is seen in (c), where the grains are colored according to an integer number that serves as an id for the grains. This number was assigned so that the smallest number corresponds to grains with the largest volume and vice versa.

Figure 5 :

 5 Figure 5: (a) Global kinetics, (b) individual kinetics and final texture (c) after 4 ns of computational annealing at 500 • C.

Figure 6 :

 6 Figure6: (a) Average grain rotation and (b) individual rotation of the grains. Grain rotation seems to increase with decreasing grain size of the grains. However, the average orientation change of the grains remains below 3.5 • . Significant events for microstructure evolution can also be identified since properties such as kinetics (c) and orientation change (d) of the grains are individually resolved. In (c) and (d), the properties plotted correspond to the grain shown Fig.7.

Figure 7 :

 7 Figure 7: Development of an arbitrary grain as reconstructed by the GraDe-A tool. The surface was calculated by means of the visualization software OVITO [41]. This grain initially grew (b) until it reached a maximum number of faces after 1.33 ns and a maximum size (c) after 2.66 ns. Afterward, the grain shrank (d) until the simulation finalized after 4 ns.

1 foreach atom a do 2 3 4 if n N N > 12 then 5 return not oriented 6 end 7

 234567 Identify all n N N nearest neighbor atoms A N N Calculate the relative positions V N N of A N N to a // The fcc-lattice has twelve neighbors, too much neighbors may cause problems.Calculate the unit vectors to V N N and save them in V N N // Find the antipodal partners and calculate the mean direction

8

 8

9 if n R < 6 then 10 return not oriented 11 end // Calculate three 100 directions 12 17 end // check all ij = ji combinations 18 for j = i + 1 ; 21 v 29 if 30 return not oriented 31 end 32 Setupforeach atom a do 2 if a has been assigned to any grain then 3 continue 4 end 5 new grain G 6 empty atom list l N 7 empty atom list l N N 8 add a to l N with a as parent atom 9 while 13 end // local criterion 14 if a n and its parent atom have not a very close orientation then 15 continue 16 end // global criterion 17 if 19 end 20 assign a n to G 21 add 23 l N = l N N 24 l N N = new empty list 25 end 26 if G does not comprise a sufficient amount of atoms then 27 dissociate all atoms of G 28 delete G 29 end

 910111217181212930322345678913141617192021232426272829 Empty 100 -vector list V 100 13 for i = 0; i < n R ; i++ do 14 v i is the ith vector in V R 15 if the number of vectors inside V 100 is three then16 break j < n R ; j++ do 19 v j is the jth vector in V R 20 if the angle between v i and v j is in the interval 90 • ± ∆β then // Calculate the crossproduct of v i with v j CP = v i × v j 22 Append v CP to V 100 23if the number of vectors inside V 100 is three then the number of vectors inside V 100 is smaller than three then the matrix M as shown in equation (3)33Calculate the orientation quaternion q as a least square solution to M , utilizing a 4 × 4-Matrix and computing the corresponding eigenvectors with Armadillo 34 return the orientation id to q 35 end Result: The orientation id to the atoms orientation Algorithm A3: Grain-identification algorithm grouping areas of similarly oriented atoms into grain entities. Data: Position data for all atoms Result: A certain number of grains 1 number of entries of l N > 0 do 10 foreach atom a n in l N do 11 if a n has been assigned to any grain then 12 continue the orientation of a n and the average orientation of G are not close then 18 continue all nearest neighbors of a n to l N N with a n as parent atom 22 end

Table 1 :

 1 Comparison of the total computation time t comp necessary for processing 24 output-files by the Grade-A tool. A different number of threads (#Th) were used for benchmarking in a linux-environment with 12 CPU-cores. Each file contained about 4.8 million atoms and had a size of approximately 240 MB. The relative overhead was calculated using the computation with two threads as reference.

	#Th t comp (h) t comp ×#Th (h) Overhead (%)
	2	1.02	2.03	0.00
	4	0.53	2.12	3.78
	6	0.37	2.23	9.469
	8	0.29	2.28	12.17
	10	0.28	2.77	36.10
	12	0.20	2.43	19.29

Acknowledgements

The authors express their gratitude to the Deutsche Forschungsgemeinschaft (DFG) for financial support (Grants GO 335/44-1) and gratefully acknowledge the computing time granted (JHPC24) by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC). P. W. Hoffrogge thankfully acknowledges the computing time granted by the RWTH-Aachen University (RWTH0060) for the execution of his master-thesis project.

Appendix A. Appendix

Appendix A.1. Per-atom orientation calculation An orientation is calculated by algorithm A1 for every atom by taking into account the positions of the nearest neighbors. During the calculation a reduced neighbor vector list V R is calculated for every atom by algorithm A2. This procedure eliminates double occurrences of anti-parallel neighbor unit-vectors by calculating the mean direction Algorithm A2: Algorithm reducing a nearest neigbor vector list by eliminating twice occurrences of antipodal vectors.

Input:

Vector list V N N containing n N N nearest neighbor unit-vectors of an atom The antipodal threshold angle ∆α

. Grain identification and reconstruction

In order to group atoms together into a grain entity, two different criteria were introduced. A local criterion that compares the orientation of a candidate atom i.e. that of the atom to be assigned to a certain grain and that of the reference atom, referred to as parent atom. A global criterion compares the average orientation of the grain and the orientation of the candidate atom.

Appendix A.3. Orphan atom adoption

The atoms that were not successfully assigned to a grain by the previous algorithm are referred to as orphan atoms. The current algorithm is used to assign these atoms to a grain in a procedure deemed as atom adoption, which is based on the most frequent occurrences of the same grain. A threshold for the minimum number of occurrences of a grain was implemented to suppress a random pick behavior. The procedure is repeated for every atom until no further adoptions can occur in the dataset.

Algorithm A4: Algorithm assigning atoms to preexisting grains by taking into account the grain-membership of nearest neighbor atoms.

Data: Position data for all atoms

Result: A certain number of grains Both the center-of-mass (COM) and the orientation of the grains are utilized as criteria to match couples from the data sets.

Algorithm A5: Algorithm assigning a currently detected grain to a grain-entity from a previous timestep.

Input: a newly identified grain g Data: Volume V G of the grain g some factor a

1 mapped grain g mapped grain id id G = NO GRAIN

1/3 Algorithm A6: Algorithm to obtain the unique cubical orientation quaternion q uo to a quaternion q.

Input: An orientation quaternion q

The unique quaternion to identify q u 3 foreach 24 cubically equivalent quaternions q i to q do

List of the 24 cubically-equivalent orientation quaternions:

q T 10 = 0.5[a 0 -a 1 + a 2 + a 3 ; a 0 + a 1 -a 2 + a 3 , -a 0 + a 1 + a 2 + a 3 , -a 0 -a 1 -a 2 + a 3 ] (A.10)

Grain reconstruction and determination of grain attributes from MD data sets of FCC polycrystals 26

[-a 2 -a 3 ; a 2 -a 3 , a 0 -a 1 , a 0 + a 1 ] (A.17)

[-a 1 -a 3 ; a 0 + a 2 , -a 1 + a 3 , a 0 -a 2 ] (A.18)

[a 0 + a 1 ; -a 0 + a 1 , a 2 -a 3 , a 2 + a 3 ] (A.19)

[a 0 + a 2 ; a 1 + a 3 , -a 0 + a 2 , -a 1 + a 3 ] (A.20)

[a 0 + a 3 ; a 1 -a 2 , a 1 + a 2 , -a 0 + a 3 ] (A.21)

[a 1 -a 2 ; -a 0 -a 3 , a 0 -a 3 , a 1 + a 2 ] (A.22)

[a 2 -a 3 ; a 2 + a 3 , -a 0 -a 1 , a 0 -a 1 ] (A.23)

[a 1 -a 3 ; -a 0 + a 2 , -a 1 -a 3 , a 0 + a 2 ] (A. 24)