On Some Grüss-Type Fractional Inequalities Using Saigo Fractional Integral Operator

The main aim of this paper is to establish some new fractional integral inequalities of Grüss-type via the Saigo fractional integral operator. 1. Introduction Grüss inequality is an inequality which establishes a connection between the integral of the product of two functions and the product of the integrals of the two functions. In 1935, Grüss proved the following well-known classical integral inequality; see [1, 2]. Theorem 1 (see [2]). Let be two integrable functions such that and for all ; , and are constant; then where the constant is sharp. In the past decade, many researchers had numerous generalizations, variants, and extensions of Grüss inequality done in literature; to mention a few, see [1–12] and the references cited therein. Recently, many authors have studied the fractional integral inequalities via Caputo, Riemann-Liouville, and -fractional integral; see [6, 13–21]. Some authors have studied the Saigo fractional integral operator; for example, we refer the reader to [22–29] and references cited therein. In [19], Dahmani et al. gave the following fractional integral inequality using Riemann-Liouville fractional integral. Theorem 2 (see [19]). Let and be two integrable functions on satisfying the condition on . Then, for all and , one has In literature few results have been obtained on some fractional integral inequalities using Hadamard fractional integral and Saigo fractional integral operator in [25, 26, 30–32]. Our purpose in this paper is to establish some new results using Saigo fractional integral. 2. Preliminaries We give some definitions and properties which will be used later. For more details, see [28, 33]. Definition 3. A real-valued function ( ) is said to be in space , if there exists a real number such that , where . Definition 4 (see [27, 28]). Let , ; then the Saigo fractional integral of order for a real-valued continuous function is defined by where the function in the right-hand side of (4) is the Gaussian hypergeometric function defined by And is the Pochhammer symbol For in (4), we have the known result [28] as . 3. Grüss-Type Fractional Integral Inequality In this section, we establish Grüss-type fractional integral inequality involving Saigo fractional integer operator (4), for which we assume the following. There exist two integrable functions and on , such that There exist two integrable functions and on , such that Lemma 5. Suppose that is an integrable function on and and are two integrable functions on . Assume that the condition holds. Then, for all , , , , one has Proof. Let be an integrable function on . For