Certain mean-type fractional integral inequalities via different convexities with applications

Muhammad Samraiz,Kottakkaran Sooppy Nisar,Gauhar Rahman,Sajid Iqbal,Thabet Abdeljawad,Fakhra Nawaz

doi:10.1186/s13660-020-02474-x

Abstract

In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for (s+1)-convex functions involving the (k,s)-Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities.

Highlights

The well known Hermite–Hadamard inequality for a convex function Ψ : U → R on an interval U of real numbers, with φ, φ ∈ U and φ < φ is given by φ+φ Ψ ≤ φ Ψ (ξ ) dξ (φ) + (φ) . (1.1) φ–φ φNumerous scientists examined this inequality and published various generalizations and extensions by using fractional integrals and derivatives [5, 8, 15, 16, 18, 19, 23, 25,26,27,28,29, 32, 33]
We give some key definitions and mathematical fundamentals of the theory of fractional calculus which are utilized in this paper
Example 2.5 By plotting the graphs of inequalities of Theorem 2.4 for the convex function ψ(℘) = ℘2 and g = 2, we prove the validity of the results

Summary

Introduction

Definition 1.1 ([14]) A function ψ : [φ, φ] → R is called convex if the following inequality holds on an interval [φ, φ] ⊆ R: ψ νl + (1 – ν)r ≤ νψ(l) + (1 – ν)ψ(r), where l, r ∈ [φ, φ], and ν ∈ [0, 1]. Definition 1.3 ([10]) The left and right conformable fractional integral operators Jφχ+,β and Jφχ–,β of order β ∈ C, such that Re(β) > 0 and 0 < χ ≤ 1, for ψ ∈ L1[φ, φ] are defined by

Results

Conclusion