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A Generalization of the Kratzel Function and Its Applications.

1. Introduction

The Kratzel function is defined for x >0 by the integral

[mathematical expression not reproducible], (1)

where [rho] [member of] R and v [member of] C, such that R(v) < 0 for [rho] [less than or equal to] 0 (cf. [1]). For [rho] [greater than or equal to] 1 the function (1) was introduced by Kratzel as a kernel of the integral transform as follows:

([K.sup.[rho].sub.v] f) (x) = [integral] [Z.sup.v.sub.[rho]] (xt) f (t) dt (x > 0). (2)

The Kratzel function [Z.sup.v.sub.[rho]](x) is related to the modified Bessel function of the second kind [K.sub.v] by the relationship

[Z.sup.v.sub.1] (x) = 2[x.sup.v/2][K.sub.v] (2[square root of x]). (3)

The generalized Kratzel function [D.sup.v,[alpha].sub.[rho],r] (x) is given in [2, 3] by the following relation:

[mathematical expression not reproducible], (4)

where [rho] [member of] R, r [member of] [R.sup.+], v [member of] C, and [alpha] > 1. Kilbas and Kumar considered the special case for r = 1 in [2], calculated fractional

derivatives and fractional integrals of [D.sup.v,[alpha].sub.[rho],1](x), and obtained a representation using Wright hypergeometric functions. On the other hand the general case of (1) is given in [2, (54), p. 845].

We consider the generalized Kratzel function [Y.sup.v.sub.[rho],r](x) defined by the integral

[mathematical expression not reproducible], (5)

for x > 0, [rho] [member of] R, r [member of] [R.sup.+], and v [member of] C. The function [Y.sup.v.sub.[rho],r](x) is a generalization of the Kratzel function [Z.sup.v.sub.[rho]] (x) since

[mathematical expression not reproducible]. (6)

If a = 1 in (4), then

[mathematical expression not reproducible]. (7)

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

[f.sub.n] (X) x [f.sub.n+2] (x) - [[[f.sub.n-1] (x)].sup.2] [greater than or equal to] 0, n = 0, 1, 2, ... (8)

are important and well known in many fields of mathematics (cf. [4]). A function f(x) is completely monotonic on (0, [infinity]), if f has derivatives of all orders and satisfies the inequality

[(-1).sup.m] [f.sup.(m)] (x) [greater than or equal to] 0 (9)

for all x > 0 and m [member of] N (cf. [5, Section IV]). A function f(x) is said to be log-convex on (0, [infinity]), if

f [[alpha][x.sub.1] + (1 - [alpha])[x.sub.2]] [less than or equal to] [[f([x.sub.1])].sup.[alpha]] [[f([x.sub.2])].sup.1-[alpha]] (10)

for all [x.sub.1], [x.sub.2] > 0 and [alpha] [member of] [0,1] (cf. [5, p. 167]).

Let p,q [member of] R such that p > 1 and 1/p + 1/q = 1 .If f and g are real valued functions defined on a closed interval and [[absolute value of (f)].sup.p], [[absolute value of (g)].sup.q] are integrable in this interval, then we have

[mathematical expression not reproducible]. (11)

The following inequality is due to Mitrinovic et al. (cf. [6, p. 239]). Let f and g be two functions which are integrable and monotonic in the same sense on [a, b] and p is a positive and integrable function on the same interval, then the following inequality holds true:

[mathematical expression not reproducible], (12)

if and only if one of the functions f and g reduces to a constant.

The Mellin transform of the function f is defined by

M {f(x); s} = [[integral].sup.[infinity].sub.0] [x.sup.s-1]f(x)dx (13)

when M{f(x); s} exists. The Mellin transform of the generalized Kratzel function (5) is given by Kilbas and Kumar in [2].

The Laplace transform of the function f is defined by

L {f(x); s} = 1/[GAMMA]([alpha]) [[integral].sup.[infinity].sub.0] [e.sup.[alpha]-1] f(x)dx (14)

provided that the integral on the right-hand side exists.

The Liouville fractional integral is defined by

([F.sup.[alpha].sub.-]f)(x) = 1/[GAMMA]([alpha]) [[integral].sup.[infinity].sub.x] [(t - x).sup.[alpha]-1] f(t) dt (15)

and its derivatives [F.sup.[alpha].sub.-] and [D.sup.[alpha].sub.-] are

[mathematical expression not reproducible], (16)

where x > 0, [alpha] [member of] C, and R([alpha]) > 0 (cf. [7, Section 5.1]).

We introduce new operators

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible], (18)

where v [member of] C and [lambda] > 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 [9-11].

In this paper, we investigate the properties of the functions [Y.sup.v.sub.[rho],r](x) and prove their composition of [Y.sup.v.sub.[rho],r](x) with fractional integral and derivatives ([J.sup.[alpha].sub.-]f)(x), ([D.sup.[alpha].sub.-]f)(x) given by (15) and (16) (cf. [2, 6,12,13]). In Section 3, we show that [Y.sup.v.sub.[rho],r](x) is the solution of differential equations of fractional order.

2. The Main Theorems

In this section, we will give some properties of generalized Kratzel functions [Y.sup.v.sub.[rho],r].

Lemma 1. Let [rho] [member of] R ([rho] [not equal to] 0), r [member of] [R.sup.+], v [member of] C, R(s) > 0 be such that R(v + rs) > 0 when [rho] > 0 and R(v + rs) < 0 when [rho] < 0. The Mellin transform of the function [Y.sup.v.sub.[rho],r](x) is given by

M{[Y.sup.v.sub.[rho],r]; s} = 1/[absolute value of ([rho])] [GAMMA](s)[GAMMA] (v + rs/[rho]) x > 0. (19)

Proof. Using (13) and (5), we have

[mathematical expression not reproducible]. (20)

Changing the order of integration and using the substitution of [xt.sup.-r] = u, we have

[mathematical expression not reproducible]. (21)

Making the change of variable the integral [t.sup.[rho]] = z, and using the known formula (1) from [14, p. 145], we find that

[mathematical expression not reproducible], (22)

when [rho] > 0 and

[mathematical expression not reproducible], (23)

when [rho] < 0.

Theorem 2. We have the following relationship for the function [Y.sup.v.sub.[rho],r](x):

[mathematical expression not reproducible], (24)

where [rho] [member of] R, r [member of] [R.sup.+], v [member of] C, and x > 0.

Proof. Using (5) and making the change of [t.sup.-r] = u, we obtain

[mathematical expression not reproducible]. (25)

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

[mathematical expression not reproducible], (26)

for [rho] = 1:

[mathematical expression not reproducible]. (27)

Theorem 3. If [rho] [member of] R, r [member of] [R.sup.+], v [member of] C and x > 0, then the following assertions are true:

(a) The function [Y.sup.v.sub.[rho],r] (x) satisfies the recurrence relation

v[Y.sup.v.sub.[rho],r] (x) = [rho][Y.sup.v+[rho].sub.[rho],r] (x) - rx[Y.sup.v-r.sub.[rho],r] (x). (28)

(b) The function x [right arrow] [Y.sup.v.sub.[rho],r] (x) is completely mono tonic on (0, [infinity]).

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

[mathematical expression not reproducible]. (29)

(b) From Bernstein-Widder theorem (see Theorem 1, [5, p. 145]), the function [Y.sup.v.sub.[rho],r](x) is completely monotonic on (0, [infinity]) for all x >0. This could be verified directly as follows:

[mathematical expression not reproducible], (30)

which follows via mathematical induction from (5) provided that [rho] [member of] R, r [member of] [R.sup.+], v [member of] C and x > 0. From Bernstein-Widder theorem, generalized forms of Kratzel function are completely monotonic on (0, rn) for all x >0. Due to (30), the functions are completely monotonic on (0, [infinity]) for all x > 0.

Setting r [right arrow] 1 and using (28), the equation yields

[mathematical expression not reproducible]. (31)

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

[mathematical expression not reproducible], (32)

(cf. 2.1 of Theorem 1 from [12]).

Theorem 4. Let [v.sub.1], [v.sub.2], [rho] [member of] R, 0 < [lambda] < 1, and x > 0, then the following assertions hold true:

(a) The function v [right arrow] [Y.sup.v.sub.[rho],r] (x) is log-convex on R:

[mathematical expression not reproducible]. (33)

(b) The function x [right arrow] [Y.sup.v.sub.[rho],r] (x) is log-convex on (0, [infinity]):

[mathematical expression not reproducible]. (34)

(c) The function [Y.sup.v.sub.[rho],r](x) satisfies the following relation:

[Y.sup.v.sub.[rho],r] ([t.sup.r]) = [t.sup.v][Y.sup.-v.sub.r,[rho]]([t.sup.[rho]]). (35)

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

[mathematical expression not reproducible], (36)

where [lambda] [member of] [0,1], [v.sub.1], [v.sub.2], [rho] [member of] R, [alpha] > 1, and x > 0. Thus, v [right arrow] [Y.sup.v.sub.[rho],r](x) is log-convex on R.

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

[mathematical expression not reproducible], (37)

where [lambda] [member of] [0,1], v, [rho] [member of] R, r > 0, and [x.sub.1], [x.sub.2] > 0. Thus, x [right arrow] [Y.sup.v.sub.[rho],r](x) is log-convex on (0, rn).

(c) Again using (5), we conclude that

[mathematical expression not reproducible] (38)

or for the change of x = [t.sup.r], we obtain (35).

Moreover, since [Y.sup.v.sub.[rho],r](x) is log-convex on R, we have Turan type inequality

[mathematical expression not reproducible] (39)

for [v.sub.1], [v.sub.2], [rho] [member of] R, [alpha] > 1, and x > 0. Making the change of variable [v.sub.1] = v - 2 and [v.sub.2] = v, the equation yields

[mathematical expression not reproducible] (40)

which is valid for v, [rho] [member of] R, [alpha] > 1, and x > 0.

Using (39) and making the change of variables [v.sub.1] = v - n - 1 and [v.sub.2] = v- n +1, we have

[mathematical expression not reproducible]. (41)

Theorem 5. If v, [rho] [member of] R, r [member of] [R.sup.+] and x > 0, then the following inequality holds true:

[mathematical expression not reproducible]. (42)

Proof. Let [mathematical expression not reproducible]. The function f(t) is increasing on (0, [infinity]) for v [greater than or equal to] 1 and is decreasing for v [less than or equal to] 1. On the other hand, we observe that, for all [rho] > 0,

g'(t)/g(t) = [rho][t.sup.-[rho]-1] > 0. (43)

Thus, g(t) is increasing if and only if [rho] > 0. Moreover, making the change of [t.sup.r] = u and using the known formula (1) from [14, p. 137], we have

[mathematical expression not reproducible]. (44)

Making the change of [t.sup.-r] = u, we find

[mathematical expression not reproducible]. (45)

Making the change of variable t = [u.sup.-1/r] and using (6), we have

[mathematical expression not reproducible]. (46)

Using (5) and making the change of variable t = [u.sup.-1], we find

[mathematical expression not reproducible]. (47)

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

[mathematical expression not reproducible]. (48)

If we choose r [right arrow] 1 in (42), then we have

[x.sup.v-1][Y.sup.-v.sub.[rho],1] (x) [greater than or equal to] [GAMMA] (v) [Y.sup.-1.sub.[rho],1] (X). (49)

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

[Z.sup.-v.sub.[rho]] (x) [greater than or equal to] [x.sup.1-v][GAMMA](v)[Z.sup.-1.sub.[rho]] (x). (50)

3. Differential Equations of Fractional Order

In this section, we show that [Y.sup.v.sub.[rho],r](x) is the solution of differential equations of fractional order.

Theorem 6. If [alpha], v [member of] C, R([alpha]) > 0, and [rho] > 0, then the following identity holds true:

([J.sup.[alpha].sub.-][Y.sup.v.sub.[rho],r](x) = [Y.sup.v+r[alpha].sub.[rho],r] (x). (51)

Proof. Applying (15), (5), and relation (11) of [15, p. 202], we obtain

[mathematical expression not reproducible]. (52)

Theorem 7. If [alpha], v [member of] C, R([alpha]) > 0, and [rho] > 0 then we have

([D.sup.[alpha].sub.-][Y.sup.v.sub.[rho],r])(x) = [Y.sup.v-r[alpha].sub.[rho],r](x). (53)

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

[mathematical expression not reproducible]. (54)

Corollary 8. If [alpha], [beta], and v [member of] C, R([alpha]) > 0, R([beta]) > 0, and [rho] > 0, then we have

[mathematical expression not reproducible]. (55)

Theorem 9. If v [member of] C and [rho] > 0, then the following identity holds true:

[L.sup.v.sub.[rho]] [Y.sup.v.sub.[rho],r] (x) = -[rho][Y.sup.v+(1-r)[rho].sub.[rho],r] (x). (56)

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

[mathematical expression not reproducible]. (57)

Using the formula

[mathematical expression not reproducible] (58)

and applying the integration by parts, we find

[mathematical expression not reproducible] (59)

Corollary 10. If v [member of] C and [rho] > 0, then the function [Y.sup.v.sub.[rho],r](x) is a solution of the differential equation of fractional order

[mathematical expression not reproducible]. (60)

Remark 11. If v [member of] C, and [rho] = r = 1, then the function [Y.sup.v.sub.l,1](x) = [Z.sup.v.sub.1](x) is a solution of the following differential equation:

xy" + (v -1) y' - y = 0 (61)

(cf. [13, (30), p. 20]).

Theorem 12. If v [member of] C and [rho] > 0, then the function [Y.sup.v.sub.[rho],r](x) is a solution of the differential equation of fractional order

[mathematical expression not reproducible]. (62)

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

[mathematical expression not reproducible]. (63)

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

[mathematical expression not reproducible]. (64)

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

[mathematical expression not reproducible]. (65)

If we rewrite the expression in (65) relation as

TX TX

V - r[rho] + rx/[t.sup.r] = v -(2r - 1)[rho] + rx/[t.sup.r] + (r - 1)[rho], (66)

then we have

[mathematical expression not reproducible]. (67)

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:

[mathematical expression not reproducible], (68)

where

[mathematical expression not reproducible]. (69)

Remark 13. If v [member of] C and [rho] = 1, then the function [Y.sup.v.sub.1,1](x) = [Z.sup.v.sub.1] (x) is a solution of the differential equation of fourth order

[x.sup.2][y.sup.(IV)] + (2v - 4) xy'" (v -1) (v -2) y" + y = 0 (70)

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

4. Conclusion

Mejer's G functions, which are generalization of hypergeometric functions, are Mellin-Barnes integrals. Generalized Kratzel functions, [Y.sup.v.sub.[rho],r](x) could be written in terms of H-functions, which are generalization of G-function, as a Mellin-Barnes integral. Furthermore, the integral transform with the kernel [Y.sup.v.sub.[rho],r](x) could be investigated.

http://dx.doi.org/ 10.1155/2017/2195152

Competing Interests

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

References

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[3] D. Kumar, "Some among generalized Kratzel function, P-transform and their applications," in Proceedings of the National Workshop on Fractional Calculus and Statistical Distributions, pp. 47-60, CMS Pala Compus, November 2009.

[4] P. Turan, "On the zeros of the polynomials of Legendre," Casopis pro Pestovam Matematiky a Fysiky, vol. 75, no. 3, pp. 113-122, 1950.

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[7] S. G. Samko, A. A. Kilbas, and O. I. M. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach S.P., New York, NY, USA, 1993.

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[9] L. Debnath, "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413-3442, 2003.

[10] F. Mainardi, Applications of Fractional Calculus in Mechanics, Transform Methods and Special Functions, Varna 96, Edited by P. Rusev, I. Dimovski, and V. Kiryakova, Bulgarian Academy of Sciences, Sofia, Bulgaria, 1998.

[11] R. K. Saxena and S. L. Kalla, "On a generalization of Kratzel function and associated inverse Gaussian distribution," Algebras, Groups, and Geometries, vol. 24, pp. 303-324, 2007

[12] A. Baricz, D. Jankov, and T. K. Pogany, "Turan type inequalities for Kratzel functions," Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 716-724, 2012.

[13] B. Bonilla, M. Rivero, J. Rodriguez, J. Trujillo, and A. A. Kilbas, "Bessel-type functions and bessel-type integral transforms on spaces Fp p and," Integral Transforms and Special Functions, vol. 8, no. 1-2,'pp. 13-30, 1999.

[14] A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms, vol. 1, McGraw-Hill, New York, NY, USA, 1954.

[15] A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms, vol. II, McGraw-Hill, New York, NY, USA, 1954.

Nese Dernek, (1) Ahmet Dernek, (1) and Osman Yurekli (2)

(1) Department of Mathematics, University of Marmara, Istanbul, Turkey

(2) Department of Mathematics, Ithaca College, Ithaca, NY 14850, USA

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

Received 26 July 2016; Revised 26 December 2016; Accepted 5 January 2017; Published 26 January 2017 Academic Editor: Ming-Sheng Liu
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Date:Jan 1, 2017
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