Sharps Bounds for Power Mean in Terms of Contraharmonic Mean

Zeid I Almuhiameed

doi:10.4236/jamp.2020.87093

Abstract

In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters α, β that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that a > b and let for 1) and a b, (t small) for 2) to determine the condition for α and β to become f(t) ≤ 0 and g(t) ≥ 0.

Highlights

We consider the below inequalities: (1.1)
The researchers attempt to find an answer as to what are the best possible parameters α, β that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out
In [1], the authors for the first time introduced power means defining the meaning of the term “representation” as determination of appointing of reference about which some function of variants would be minimum

Summary

Introduction

The main objective of this research work is to present optimization of inequality in the one-parameter, arithmetic and harmonic means as follows: C α a + (1−α )b,αb + (1−α ) a ≤ M p (a,b) ≤ C β a + (1− β )b, βb + (1− β ) a with C (a,b) = a2 + b2 and a+b In [1], the authors for the first time introduced power means defining the meaning of the term “representation” as determination of appointing of reference about which some function of variants would be minimum.

Results

Conclusion