^{1, 2, 3}

^{4}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

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.

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 [

In a recent paper, Purohit and Raina [

In the sequel, we use the following definitions and related details.

Two functions

A real-valued function

Let

The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the generalized fractional integral operators [

Our results in this section are based on the preliminary assertions giving composition formula of fractional integral (

Let

To prove (

Now, we obtain certain integral inequalities for the synchronous functions involving the generalized fractional integral operator (

Let

Let

Multiplying both sides of (

The following results give a generalization of Theorem

Let

To prove the above theorem, we use inequality (

It may be noted that inequalities (

For

Let

We prove this theorem by mathematical induction. Clearly, for

By the induction principle, we suppose that the inequality

Now

Making use of (

Now, we consider another variation of the fractional integral inequalities.

Let

Consider the function

Let

By applying the similar procedure as of Theorem

We now briefly consider some consequences of the results derived in the previous section. Following Curiel and Galu

Now, if we consider

Again, for

Let

Let

Indeed, by suitably specializing the values of parameters

Let

Let

Finally, if we take

We conclude with the remark that the results derived in this paper are general in character and give some contributions to the theory 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.

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