A Generalization of the Krätzel Function and Its Applications

Copyright © 2017 Neşe Dernek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce new functions Y] ρ,r(x) as a generalization of the Krätzel function. We investigate recurrence relations, Mellin transform, fractional derivatives, and integral of the function Y] ρ,r(x). We show that the function Y] ρ,r(x) is the solution of differential equations of fractional order.


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Journal of Mathematics for all  > 0 and  ∈ N (cf.[5, Section IV]).A function () is said to be log-convex on (0, ∞), if 1− (10) for all  1 ,  2 > 0 and  ∈ [0, 1] (cf.[5, p. 167]).Let ,  ∈ R such that  > 1 and 1/ + 1/ = 1.If  and  are real valued functions defined on a closed interval and ||  , ||  are integrable in this interval, then we have The following inequality is due to Mitrinović et al. (cf.[6, p. 239]).Let  and  be two functions which are integrable and monotonic in the same sense on [, ] and  is a positive and integrable function on the same interval, then the following inequality holds true: if and only if one of the functions  and  reduces to a constant.The Mellin transform of the function  is defined by when M{(); } exists.The Mellin transform of the generalized Krätzel function ( 5) is given by Kilbas and Kumar in [2].The Laplace transform of the function  is defined by provided that the integral on the right-hand side exists.The Liouville fractional integral is defined by and its derivatives I  − and D  − are where  > 0,  ∈ C, and R() > 0 (cf.[7, Section 5.1]).We introduce new operators where ] ∈ C and  > 0.
A standard source in the theory of fractional calculus is the book [8].For applications of fractional calculus to science and engineering, we refer the reader to the articles [9][10][11].

The Main Theorems
In this section, we will give some properties of generalized Krätzel functions  ] , .
> 0 be such that R(] + ) > 0 when  > 0 and R(] + ) < 0 when  < 0. The Mellin transform of the function  ] , () is given by Proof.Using ( 13) and ( 5), we have Changing the order of integration and using the substitution of  − = , we have Making the change of variable the integral   = , and using the known formula (1) from [14, p. 145], we find that when  > 0 and when  < 0.
Proof.Using (5) and making the change of  − = , we obtain Now the assertion (24) follows from the definition ( 14) of the Laplace transform.
Using the known formula (29) from [14, p. 146], we find that for  = 1: Proof.(a) The above recurrence relation could be verified by using integration by parts as follows: , () is completely monotonic on (0, ∞) for all  > 0. This could be verified directly as follows: which follows via mathematical induction from (5) provided that  ∈ R,  ∈ R + , ] ∈ C and  > 0. From Bernstein-Widder theorem, generalized forms of Krätzel function are completely monotonic on (0, ∞) for all  > 0. Due to (30), the functions are completely monotonic on (0, ∞) for all  > 0.
Using (39) and making the change of variables Theorem 5.If ],  ∈ R,  ∈ R + and  > 0, then the following inequality holds true: Proof.Let () =  −  , () =  ]−1 and () =  − − .The function () is increasing on (0, ∞) for ] ≥ 1 and is decreasing for ] ≤ 1.On the other hand, we observe that, for all  > 0, Thus, () is increasing if and only if  > 0.Moreover, making the change of   =  and using the known formula (1) from [14, p. 137], we have Making the change of  − = , we find Making the change of variable  =  −1/ and using (6), we have Using ( 5) and making the change of variable  =  −1 , we find Finally, by using the relation (12), we obtain the inequality (42): If we choose  → 1 in (42), then we have As a result, we find the following inequality by using (6):