Abstract
The aim of this paper is to establish new generalized fractional versions of the Hadamard and the Fejér–Hadamard integral inequalities for harmonically convex functions. Fractional integral operators involving an extended generalized Mittag-Leffler function which are further generalized via a monotone increasing function are utilized to get these generalized fractional versions. The results of this paper give several consequent fractional inequalities for harmonically convex functions for known fractional integral operators deducible from utilized generalized fractional integral operators.
Highlights
Fractional integral inequalities are generalizations of classical integral inequalities
The main objective of this paper is to prove some new fractional generalizations of Hadamard and the Fejér–Hadamard inequalities for harmonically convex functions
We begin with fractional integral operators defined by Salim and Faraj in [35] containing generalized Mittag-Leffler function in their kernels as follows: Definition 1 ([35]) Let σ, τ, k, r, ρ be positive real numbers and ω ∈ R
Summary
Fractional integral inequalities are generalizations of classical integral inequalities. Hadamard and Fejér–Hadamard inequalities are the inequalities which have been studied extensively for different fractional integral/derivative operators, see [1, 4,5,6, 8,9,10, 14, 16, 17, 23, 25, 27, 30, 33, 34, 36,37,38, 42, 44]. We begin with fractional integral operators defined by Salim and Faraj in [35] containing generalized Mittag-Leffler function in their kernels as follows: Definition 1 ([35]) Let σ , τ , k, r, ρ be positive real numbers and ω ∈ R. The generalized fractional integral operators containing Mittag-Leffler function for a real-valued continuous function f are defined by x ρ,r,k σ ,τ ,δ,ω,a+ f (x) =.
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