Certain Inequalities Involving Generalized Erdélyi-Kober Fractional -Integral Operators

Abstract

In recent years, a remarkably large number of inequalities involving the fractional -integral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractional -integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional -integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.

1. Introduction

Let us start by considering the following functional (see [1]): where are two integrable functions on and and are positive integrable functions on . If and are synchronous on , that is, for any , then we have (see, e.g., [2, 3]) The inequality in (2) is reversed if and are asynchronous on ; that is, for any . If for any , we get the Chebyshev inequality (see [1]). Ostrowski [4] established the following generalization of the Chebyshev inequality.

If and are two differentiable and synchronous functions on and is a positive integrable function on with and for , then we have

If and are asynchronous on , then we have If and are two differentiable functions on with and for and is a positive integrable function on , then we have Here, it is worth mentioning that the functional (1) has attracted many researchers’ attention mainly due to diverse applications in numerical quadrature, transform theory, probability, and statistical problems. Among those applications, the functional (1) has also been employed to yield a number of integral inequalities (see, e.g., [5–11]).

The study of the fractional integral and fractional -integral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional -integral operators (see, e.g., [12–16]).

The purpose of this paper is to find -calculus analogs of some classical integral inequalities. In particular, we will find -generalizations of the Chebyshev integral inequalities by using the generalized Erdélyi-Kober fractional -integral operator introduced by Galué [17]. The main objective of this paper is to present some new fractional -integral inequalities involving the generalized Erdélyi-Kober fractional -integral operator. We consider the case of synchronous functions as well as the case of functions bounded by integrable functions. Some of the known and new results are as follows, as special cases of our main findings. We emphasize that the results derived in this paper are more generalized results rather than similar published results because we established all results by using the generalized Erdélyi-Kober fractional -integral operator. Our results are general in character and give some contributions to the theory -integral inequalities and fractional calculus.

2. Preliminaries

In the sequel, we required the following well-known results to establish our main results in the present paper. The -shifted factorial is defined by where and it is assumed that .

The -shifted factorial for negative subscript is defined by We also write It follows from (8), (9), and (10) that which can be extended to as follows: where the principal value of is taken.

We begin by noting that F. J. Jackson was the first to develop -calculus in a systematic way. For , the -derivative of a continuous function on is defined by and . It is noted that if is differentiable.

The function is a -antiderivative of if . It is denoted by

The Jackson integral of is thus defined, formally, by which can be easily generalized as follows:

Suppose that . The definite -integral is defined as follows:

A more general version of (18) is given by

The classical Gamma function (see, e.g., [18, Section  1.1]) was found by Euler while he was trying to extend the factorial to real numbers. The -factorial function of defined by can be rewritten as follows: Replacing by in (22), Jackson [19] defined the -Gamma function by

The -analogue of is defined by the polynomial More generally, if , then

Definition 1. Let and . Then a generalized Erdélyi-Kober fractional integral for a real-valued continuous function is defined by (see, [17])

Definition 2. A -analogue of the Kober fractional integral operator is given by (see, [20])

Remark 3. It is easy to see that for all and . If is a continuous function, then we conclude that, under the given conditions in (26), each term in the series of generalized Erdélyi-Kober -integral operator is nonnegative and thus for all and .
On the same way each term in the series of Kober -integral operator (27) is also nonnegative and thus for all and .

3. Inequalities Involving a Generalized Erdélyi-Kober Fractional -Integral Operator for Synchronous Functions

This section begins by presenting two inequalities involving generalized Erdélyi-Kober -integral operator (26) stated in Lemmas 4 and 5 below.

Lemma 4. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Proof. Let and be two continuous and synchronous functions on . Then, for all , with , we have or, equivalently,
Now, multiplying both sides of (33) by , integrating the resulting inequality with respect to from to , and using (26), we get
Next, multiplying both sides of (34) by , integrating the resulting inequality with respect to from to , and using (26), we are led to the desired result (31).

Lemma 5. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Proof. Multiplying both sides of (34) by which remains nonnegative under the conditions in (35), integrating the resulting inequality with respect to from to , and using (26), we get the desired result (35).

Theorem 6. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Proof. By setting and in Lemma 4, we get Since under the given conditions, multiplying both sides of (38) by , we have Similarly replacing , by , and , by , , respectively, in (31) and then multiplying both sides of the resulting inequalities by and both of which are nonnegative under the given assumptions, respectively, we get the following inequalities: Finally, by adding (39), (40), and (41), side by side, we arrive at the desired result (37).

Theorem 7. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Proof. Setting and in (35), we have Multiplying both sides of (43) by , after a little simplification, we get Now, by replacing , by , and , by , in (35), respectively, and then multiplying both sides of the resulting inequalities by and , respectively, we get the following two inequalities: Finally, we find that the inequality (42) follows by adding the inequalities (44) and (45), side by side.

Remark 8. It may be noted that inequalities (37) and (42) in Theorems 6 and 7, respectively, are reversed if the functions are asynchronous on . The special case of (42) in Theorem 7 when , , and is easily seen to yield inequality (37) in Theorem 6.

Remark 9. We remark further that we can present a large number of special cases of our main inequalities in Theorems 6 and 7. Here, we give only two examples: setting in (37) and in (42), we obtain interesting inequalities involving Erdélyi-Kober fractional integral operator.

Corollary 10. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Corollary 11. Let , let and be two continuous and synchronous functions on , and let be continuous functions. Then, the following inequality holds true: for all and .

Remark 12. If we take and in Theorem 6 and and in Theorem 7, then we obtain the known results due to Dahmani [21].

4. Inequalities Involving a Generalized Erdélyi-Kober Fractional -Integral Operator for Bounded Functions

In this section we obtain some new inequalities involving Erdélyi-Kober fractional -integral operator in the case where the functions are bounded by integrable functions and are not necessary increasing or decreasing as are the synchronous functions.

Theorem 13. Let , let be an integrable function on , and let be continuous functions. Assume the following.There exist two integrable functions on such that Then, for , , and , we have

Proof. From , for all and , we have Therefore, Multiplying both sides of (51) by , , and integrating both sides with respect to on , we obtain Multiplying both sides of (52) by , , and integrating both sides with respect to on , we get inequality (49) as requested. This completes the proof.

As special cases of Theorems 13, we obtain the following results.

Corollary 14. Let , let be an integrable function on satisfying , for all , let be continuous functions, and let  . Then, for , , and , we have

Corollary 15. Let , let be an integrable function on , and let be continuous functions. Assume that there exists an integrable function on and a constant such that for all , , and ; we have

Theorem 16. Let , let be an integrable function on , let be continuous functions, and let satisfying . Suppose that holds. Then, for , , and , we have

Proof. According to the well-known Young inequality [3] and setting and , , we have Multiplying both sides of (58) by for , and integrating with respect to and from to , we deduce the desired result in (56).

Corollary 17. Let , let be an integrable function on satisfying , for all , let be continuous functions, and let . Then, for , , and , we have

Theorem 18. Let , let be an integrable function on , let be continuous functions, and let satisfying . In addition, suppose that holds. Then, for , , and , we have

Proof. From the well-known Weighted AM-GM inequality [3] by setting and , , we have Multiplying both sides of (63) by for , and integrating with respect to and from to , we deduce inequality (61).

Corollary 19. Let , let be an integrable function on satisfying , for all , let be continuous functions, and let . Then, for , , and , we have

Lemma 20 (see [22]). Assume that , , and . Then,

Theorem 21. Let , let be an integrable function on , let be a continuous function, and let constants , . In addition, assume that holds. Then, for any , , , and , the following two inequalities hold:

Proof. By condition and Lemma 20, for , , it follows that for any . Multiplying both sides of (69) by , , and integrating the resulting identity with respect to from to , one has inequality . Inequality is proved by setting in Lemma 20.

Corollary 22. Let , let be an integrable function on satisfying , for all , let be continuous functions, and let . Then, for , , and , we have

Theorem 23. Let , let and be two integrable functions on , and let be continuous functions. Suppose that holds and moreover we assume the following.  There exist and integrable functions on such that Then, for , , and , the following inequalities hold:

Proof. To prove , from and , we have for that Therefore, Multiplying both sides of (74) by , , and integrating both sides with respect to on , we obtain Multiplying both sides of (75) by , , and integrating both sides with respect to on , we get the desired inequality .
To prove –, we use the following inequalities:

Theorem 24. Let and be two integrable functions on , let be continuous functions, and let satisfying . Suppose that and hold. Then, for , , and , the following inequalities hold:

Proof. The inequalities – can be proved by choosing the parameters in the Young inequality [3]:

Theorem 25. Let and be two integrable functions on , let be continuous functions, and let satisfying . Suppose that and hold. Then, for , , and , the following inequalities hold:

Proof. The inequalities – can be proved by choosing the parameters in the Weighted AM-GM [3]:

Theorem 26. Let and be two integrable functions on , let be continuous functions, and let constants , . Assume that and hold. Then, for any , , , and , the following inequalities hold:

Proof. The inequalities – can be proved by choosing the parameters in Lemma 20:

5. Concluding Remark

We conclude our present investigation with the remark that the results derived in this paper are general in character and give some contributions to the theory of -integral inequalities and fractional calculus. Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. In last, use of the generalized Erdélyi-Kober fractional -integral operator due to Gaulué is the advantage of our results because after setting suitable parameter values in our main results, we get known results established by number of authors.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.