Abstract
By Hölder’s integral inequality, the authors establish some Hermite–Hadamard type integral inequalities for n-times differentiable and geometrically quasi-convex functions.
Highlights
IntroductionA function f : I → R is said to be convex if f ( x + (1 − )y) ≤ f (x) + (1 − )f (y) (1)
Let I be an interval on R = (−∞, ∞)
The aim of this paper is to find more integral inequalities of Hermite–Hadamard type for n-times differentiable and geometrically quasi-convex functions
Summary
A function f : I → R is said to be convex if f ( x + (1 − )y) ≤ f (x) + (1 − )f (y) (1). If the inequality (1) reverses, f is said to be concave on I. A function f : I ⊆ R+ = (0, ∞) → R+ is said to be geometrically convex on I if f x y1− ≤ [f (x)] [f (y)]1−. One of the most famous inequalities for convex functions is Hermite–Hadamard’s inequality: if f : I ⊆ R → R is convex on an interval I of real numbers and a, b ∈ I with a < b, a+b f ≤ b f (x)dx f (a) +. If f is concave on I, the inequality (2) is reversed. We collect several Hermite–Hadamard type integral inequalities as follows
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