Abstract
In this paper we present some inequalities of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex. Moreover, an application to special means of real numbers is also considered.
Highlights
A real-valued function f defined on an interval I ⊆ R is said to be convex on I, if f ( x + (1 − )y) ≤ f (x) + (1 − )f (y) for all x, y ∈ I and ∈ [0, 1]
The main purpose of this paper is to present a parametrized inequality of HermiteHadamard type for functions whose third derivative absolute values are quasi-convex
We present a parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex, the main results are given in Theorems 3.1 and 3.3
Summary
A real-valued function f defined on an interval I ⊆ R is said to be convex on I, if f ( x + (1 − )y) ≤ f (x) + (1 − )f (y) for all x, y ∈ I and ∈ [0, 1]. If f is convex on I, we have the Hermite-Hadamard inequality (see Mitrinović et al 1993). A function f : I ⊆ R → R is said to be quasi-convex on I, if f ( x + (1 − )y) ≤ max f (x), f (y) for all x, y ∈ I and ∈ [0, 1]. Any convex function is a quasi-convex function. In 2007, Ion (2007) presented an inequality of Hermite-Hadamard type for functions whose derivatives in absolute values are quasi-convex functions, as follows: Wu et al SpringerPlus (2015) 4:831
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