Research Article | Open Access
Neşe Dernek, Ahmet Dernek, Osman Yürekli, "A Generalization of the Krätzel Function and Its Applications", Journal of Mathematics, vol. 2017, Article ID 2195152, 7 pages, 2017. https://doi.org/10.1155/2017/2195152
A Generalization of the Krätzel Function and Its Applications
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.
The Krätzel function is defined for by the integralwhere and , such that for (cf. ). 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 relationshipThe generalized Krätzel function is given in [2, 3] by the following relation:where , , , and . Kilbas and Kumar considered the special case for in , 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 integralfor , , , and . The function is a generalization of the Krätzel function sinceIf 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. ). A function is completely monotonic on , if has derivatives of all orders and satisfies the inequalityfor all and (cf. [5, Section IV]). A function is said to be log-convex on , iffor 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 Laplace transform of the function is defined byprovided that the integral on the right-hand side exists.
The Liouville fractional integral is defined byand its derivatives and arewhere , , and (cf. [7, Section 5.1]).
We introduce new operatorswhere and .
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 thatwhen andwhen .
Theorem 2. We have the following relationship for the function :where , , , and .
Using the known formula from [14, p. 146], we find thatfor :
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 .
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 obtainwhere , , , 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 inequalityfor , , and . Making the change of variable and , the equation yieldswhich 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 haveMaking the change of , we findMaking 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):
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:
Theorem 7. If , , and then we have
Corollary 8. If , and , , , and , then we have
Theorem 9. If and , then the following identity holds true:
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 getIf we take the derivative as the proof of Theorem 9, then we arrive atSubstituting (64), (16), into (63) and applying the integration by parts, we getIf we rewrite the expression in (65) relation as then we haveIf 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]).
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.
The authors declare that they have no conflict of interests regarding the publication of this paper.
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