On Fractional Integral Inequalities Involving Hypergeometric Operators

Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results. 1. Introduction Fractional integral inequalities have many applications; the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. Further, they also provide upper and lower bounds to the solutions of the above equations. These considerations have led various researchers in the field of integral inequalities to explore certain extensions and generalizations by involving fractional calculus operators. One may, for instance, refer to such type of works in the book [1] and the papers [2–11]. In a recent paper, Purohit and Raina [9] investigated certain Chebyshev type [12] integral inequalities involving the Saigo fractional integral operators and also established the -extensions of the main results. The aim of this paper is to establish certain generalized integral inequalities for synchronous functions that are related to the Chebyshev functional using the fractional hypergeometric operator, introduced by Curiel and Galu [13]. Results due to Purohit and Raina [9] and Belarbi and Dahmani [2] follow as special cases of our results. In the sequel, we use the following definitions and related details. Definition 1. Two functions and are said to be synchronous on , if for any . Definition 2. A real-valued function is said to be in the space , if there exists a real number such that , where . Definition 3. Let , , and ; then a generalized fractional integral (in terms of the Gauss hypergeometric function) of order for a real-valued continuous function is defined by [13] (see also [14]): where the function appearing as a kernel for the operator (2) is the Gaussian hypergeometric function defined by and is the Pochhammer symbol: The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the generalized fractional integral operators [13] which involves in the kernel, the Gauss hypergeometric function (defined above). The concluding section gives some special cases of the main results. 2. Main Results Our results in this section are based on the preliminary assertions giving