Some Inequalities for N-Times Differentiable s-Convex and tgs-Convex Functions

Duygu Dönmez Demir,Gülsüm Şanal

doi:10.13189/ujam.2020.080101

Abstract

The aim of this study is to present some inequalities for n-times differentiable functions. These inequalities are associated with the perturbed trapezoid inequality. n th derivatives of absolute values of the considered functions are s-convex and tgs-convex. In previous studies, some inequalities for the classes of twice differentiable functions which are convex, s-convex and tgs-convex have been introduced. We expand twice differentiable functions to n-times differentiable functions. Finally, some applications are given to verify new inequalities proposed in this study.

Highlights

Dönmez Demir and Şanal have introduced some perturbed trapezoid inequalities for n -times differentiable convex functions
We suggest some new inequalities for the classes of n-times differentiable functions which are s -convex and tgs-convex
Definition 1.2: [15] Let φ be a tgs-convex function defined by φ : I → φ (tx + (1− t ) y) ≤ t (1− t ) φ ( x) +φ ( y) (4)

Summary

Introduction

Holds for all x, y ∈[0, ∞) , t ∈[0,1] and for some constant s ∈ Definition 1.2: [15] Let φ be a tgs-convex function defined by φ : I → φ (tx + (1− t ) y) ≤ t (1− t ) φ ( x) +φ ( y) (4). Is an s -convex function in the second sense where s ∈ (0,1] and a,b ∈[0, ∞) , a < b. Theorem 1.2 (Grüss inequality): [16] Suppose φ,γ : [a,b] → are integrable, k ≤ φ ( x) ≤ l and m ≤ γ ( x) ≤ n k,l, m, n ∈ are constants), for all ∀x ∈[a,b]. + φ (n) tb + (1− t ) a dt Theorem 1.3 (Minkowski Inequality): [3] Let φ p , γ p and (φ + γ )p be integrable functions on [a,b]. We introduce some results related to the perturbed trapezoid inequality and apply them for some special means for real numbers

Main Results

2.5: Let φ :I

Some Applications