Journal of Mathematics
Volume 2017 (2017), Article ID 2195152, 7 pages
https://doi.org/10.1155/2017/2195152

Research Article

A Generalization of the Krätzel Function and Its Applications

Neşe Dernek,¹ Ahmet Dernek,¹ and Osman Yürekli²

¹Department of Mathematics, University of Marmara, Istanbul, Turkey
²Department of Mathematics, Ithaca College, Ithaca, NY 14850, USA

Correspondence should be addressed to Osman Yürekli; yurekli@ithaca.edu

Received 26 July 2016; Revised 26 December 2016; Accepted 5 January 2017; Published 26 January 2017

Academic Editor: Ming-Sheng Liu

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.

Abstract.

In this paper, we introduce new functions as a generalization of the Krätzel function. We investigate recurrence relations, Mellin transform, fractional derivatives, and integral of the function . We show that the function is the solution of differential equations of fractional order.

1. Introduction

The Krätzel function is defined for by the integral

where and , such that for (cf. [1]). For the function (1) was introduced by Krätzel as a kernel of the integral transform as follows:

The Krätzel function is related to the modified Bessel function of the second kind by the relationship

The generalized Krätzel function is given in [2, 3] by the following relation:

where , , , and . Kilbas and Kumar considered the special case for in [2], calculated fractional derivatives and fractional integrals of , and obtained a representation using Wright hypergeometric functions. On the other hand the general case of (1) is given in [2, , p. 845].

We consider the generalized Krätzel function defined by the integral

for , , , and . The function is a generalization of the Krätzel function since

If in (4), then

We give some definitions and inequalities that will be needed. The Turán type inequalities

are important and well known in many fields of mathematics (cf. [4]). A function is completely monotonic on , if has derivatives of all orders and satisfies the inequality

for all and (cf. [5, Section  IV]). A function is said to be log-convex on , if

for all and (cf. [5, p. 167]).

Let such that and . 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 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 and are

where , , and (cf. [7, Section  5.1]).

We introduce new operators

where and .

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 [9–11].

In this paper, we investigate the properties of the functions and prove their composition of with fractional integral and derivatives , given by (15) and (16) (cf. [2, 6, 12, 13]). In Section 3, we show that is the solution of differential equations of fractional order.

2. The Main Theorems

In this section, we will give some properties of generalized Krätzel functions .

Lemma 1. Let , , , be such that when and when . 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 from [14, p. 145], we find that

when and

when .

Theorem 2. We have the following relationship for the function :

where , , , and .

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 from [14, p. 146], we find that

for :

Theorem 3. If , , and , then the following assertions are true:

(a)The function satisfies the recurrence relation

(b)The function is completely monotonic on .

Proof. (a) The above recurrence relation could be verified by using integration by parts as follows:

(b) From Bernstein-Widder theorem (see Theorem  1, [5, p. 145]), the function is completely monotonic on for all . This could be verified directly as follows:

which follows via mathematical induction from (5) provided that , , and . From Bernstein-Widder theorem, generalized forms of Krätzel function are completely monotonic on for all . Due to (30), the functions are completely monotonic on for all .

Setting and using (28), the equation yields

Then using (31) and (6), we obtain the relation

(cf. 2.1 of Theorem  1 from [12]).

Theorem 4. Let , , and , then the following assertions hold true:

(a)The function is log-convex on :

(b)The function is log-convex on :

(c)The function satisfies the following relation:

Proof. (a) Using (5) and (11), we obtain

where , , , and . Thus, is log-convex on .
(b) The integrand in (5) is a log-linear convex function of . By using (11), we have

where , , , and . Thus, is log-convex on .
(c) Again using (5), we conclude that

or for the change of , we obtain (35).
Moreover, since is log-convex on , we have Turán type inequality

for , , and . Making the change of variable and , the equation yields

which is valid for , , and .
Using (39) and making the change of variables and , we have

Theorem 5. If , and , then the following inequality holds true:

Proof. Let , and . The function is increasing on for and is decreasing for . On the other hand, we observe that, for all ,

Thus, is increasing if and only if . Moreover, making the change of and using the known formula from [14, p. 137], we have

Making the change of , we find

Making the change of variable and using (6), we have

Using (5) and making the change of variable , we find

Finally, by using the relation (12), we obtain the inequality (42):

If we choose in (42), then we have

As a result, we find the following inequality by using (6):

3. Differential Equations of Fractional Order

In this section, we show that is the solution of differential equations of fractional order.

Theorem 6. If ,   , and , then the following identity holds true:

Proof. Applying (15), (5), and relation of [15, p. 202], we obtain

Theorem 7. If , , and then we have

Proof. Using (16), (5), and (51), we obtain

Corollary 8. If , and , , , and , then we have

Theorem 9. If and , then the following identity holds true:

Proof. Applying (17) to (5), we get

Using the formula

and applying the integration by parts, we find

Corollary 10. If and , then the function is a solution of the differential equation of fractional order

Remark 11. If , and , then the function is a solution of the following differential equation:

(cf. [13, , p. 20]).

Theorem 12. If and , then the function is a solution of the differential equation of fractional order

Proof. Using (18), (5), and (53), we get

If we take the derivative as the proof of Theorem 9, then we arrive at

Substituting (64), (16), into (63) and applying the integration by parts, we get

If we rewrite the expression in (65) relation as

then we have

If we evaluate the integral on the right-hand side of relation (64) and apply the integration by parts, we arrive at (62) as follows:

where

Remark 13. If and , then the function is a solution of the differential equation of fourth order

(cf. [13, p. 21]).

4. Conclusion

Mejer’s functions, which are generalization of hypergeometric functions, are Mellin-Barnes integrals. Generalized Krätzel functions, could be written in terms of -functions, which are generalization of -function, as a Mellin-Barnes integral. Furthermore, the integral transform with the kernel could be investigated.

Competing Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

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