Some New Beesack–Wirtinger-Type Inequalities Pertaining to Different Kinds of Convex Functions

Artion Kashuri,Gauhar Rahman,Muhammad Samraiz,Zareen A Khan

doi:10.3390/math10050757

Abstract

In this paper, the authors established several new inequalities of the Beesack–Wirtinger type for different kinds of differentiable convex functions. Furthermore, we generalized our results for functions that are n-times differentiable convex. Finally, many interesting Ostrowski- and Chebyshev-type inequalities are given as well.

Highlights

Introduction and PreliminariesThe following inequality regarding square integrable functions is known as the Wirtinger inequality: Theorem 1 ([1,2])
Motivated by the above results, the aim of this paper was to derive some new inequalities of the Beesack–Wirtinger type for different kinds of differentiable convex functions
Since the different kinds of convex functions that we used to obtain our results have large applications in many mathematical areas, they can be applied to derive several new important results in convex analysis, quantum mechanics, and related optimization theory and may stimulate further research in different areas of pure and applied sciences

Summary

Introduction

The following inequality regarding square integrable functions is known as the Wirtinger inequality: Theorem 1 ([1,2]). Let ∈ L2[0, 2π], : 2π be a real-valued function with period 2π and (μ)dμ = 0. With equality holding iff (μ) = A1 cos μ + A2 sin μ, where A1, A2 ∈ R. For recently published papers of this type, see [3–5]. Beesack in [6,7] generalized (1) as follows: Theorem 2.

Objectives

Results

Conclusion