Some Grüss type inequalities involving generalized fractional operator

Abstract

The analogous version of Grüss inequalities has been established using the generalized hypergeometric function fractional integral operators. The results are generalizations of Grüss type inequalities in fractional integral operators. Our main deduction will break into results noted for appropriate changes of fractional integral parameter and degree of fractional operator.

Fractional integral inequalities involving Marichev–Saigo–Maeda fractional integral operator

“…Theorem 2.1 (see [30]) If ν, ν , ξ , ξ , η ∈ R are such that η > max{ν, ν , ξ , ξ } > 0, then we have the inequality…”

On new fractional integral inequalities using Marichev–Saigo–Maeda operator

“…Definition 3. Let 𝜇, 𝜇 ′ , 𝛿, 𝛿 ′ , 𝜉 ∈ C, and R(𝜉) > max[0, R(𝜇, 𝜇 ′ , 𝛿, 𝛿 ′ )], then the following inequality holds (see Joshi et al 10 ):…”

“…Also in literature various generalizations of Grüss inequality have been obtained using different operators (see details in Dragomir, 8 Dragomir, 9 Joshi et al, 10 Kalla and Rao, 11 and Pachpatte 12 ).…”

“…Definition Let $$$ \mu, {\mu}^{\prime },\delta, {\delta}^{\prime },\xi \in C $$$, and $$$ R\left(\xi \right)>\mathit{\max}\left[0,R\left(\mu, {\mu}^{\prime },\delta, {\delta}^{\prime}\right)\right] $$$, then the following inequality holds (see Joshi et al 10 ): $$$$ {F}_3\left(\mu, {\mu}^{\prime },\delta, {\delta}^{\prime },\xi, 1-\frac{t}{x},1-\frac{x}{t}\right)>0, $$$$ provided $$$ -1<1-\frac{t}{x}<0 $$$ and $$$ 0<1-\frac{x}{t}<\frac{1}{2} $$$; also, if $$$ f(x)>0 $$$, then $$$$ {\mathcal{I}}_x^{\mu, {\mu}^{\prime },\delta, {\delta}^{\prime },\xi }f(x)&a...$$…”

Certain Generalized Fractional Integral Inequalities

“…They often describe upper and lower limits for the solutions to problems with fractional boundary value. These implications, involving fractional calculus operators, have directed numerous studies in the context of integral inequalities to investigate other extensions and generalizations [12,18,19,20,22,26]. For a comprehensive overview of the dierent applications of fractional integral inequalities, attention can be made to [1,2,3,5,6,10,14,15,16,17,28] as well as the references listed in it.…”