Fractional Minkowski-Type Integral Inequalities via the Unified Generalized Fractional Integral Operator

Abstract

This paper is aimed at presenting the unified integral operator in its generalized form utilizing the unified Mittag-Leffler function in its kernel. We prove the boundedness of this newly defined operator. A fractional integral operator comprising a unified Mittag-Leffler function is used to establish further Minkowski-type integral inequalities. Several related fractional integral inequalities that have recently been published in various articles can be inferred.

Further Generalizations of Some Fractional Integral Inequalities

“…For example, the two-parameter Mittag-Leffler function defined in [18], three-parameter Mittag-Leffler function defined in [19] and the extended Mittag-Leffler function defined in [20] can be deduced from the unified Mittag-Leffler function (2). Operators involving the unified Mittag-Leffler function are given in [21] and are defined as follows:…”

Some Fractional Integral Inequalities Involving Mittag‐Kernels

“…Multiplying both sides by the quantity given in (19) and integrating over [u, ξ], the above inequality takes the following form:…”

“…e following inequality follows by multiplying the above inequality with the quantity given in (19) and integrating over [u, ξ]: λ,ρ,θ,k,n u + ,α,β,c,δ,μ,] ϕ − fψ…”

“…Taking power m of the above inequality and multiplying by the quantity given in (19), and integrating over [u, ξ], the following inequality is obtained:…”

“…We aim to develop integral versions of Minkowski-type inequalities. We employ the generalized integral operator in collaboration with the unified Mittag-Leffler function [18,19]. Various fractional integral versions of Minkowskitype inequalities can be deduced from the main results of this paper.…”