Abstract
In current continuation, we have incorporated the notion of s- ( {alpha,m} ) -convex functions and have established new integral inequalities. In order to generalize Hermite–Hadamard-type inequalities, some new integral inequalities of Hermite–Hadamard and Simpson type using s- ( {alpha,m} ) -convex function via Riemann–Liouville fractional integrals are obtained that reproduce the results presented by (Appl. Math. Lett. 11(5):91–95, 1998; Comput. Math. Appl. 47(2–3): 207–216, 2004; J. Inequal. Appl. 2013:158, 2013). Applications to special means are also provided.
Highlights
1 Introduction Let R be the set of real numbers, I ⊆ R be an interval, and η : I ⊂ R → R be a convex in the classical sense function which satisfies the inequality η κs1 + (1 – κ)s2 ≤ κη(s1) + (1 – κ)η(s2)
The aim of this paper is to build up Hermite–Hadamard-type inequalities for Riemann– Liouville fractional integral using the s–(α, m) convexity, as well as concavity, for functions whose absolute values of the first derivative are convex
Proof Using the concavity of |f |q and the power-mean inequality, we obtain f λa + (1 – λ)b q > λ f (a) q + (1 – λ) f (b) q ≥ λ f (a) + (1 – λ)f (b)| q
Summary
Let R be the set of real numbers, I ⊆ R be an interval, and η : I ⊂ R → R be a convex in the classical sense function which satisfies the inequality η κs1 + (1 – κ)s2 ≤ κη(s1) + (1 – κ)η(s2)whenever s1, s2 ∈ I and κ ∈ [0, 1]. If |η | is a convex function on [s1, s2], the following inequality holds: s2 η(u) du – η s1 + s2
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