Some New Inequalities Between Important Means

TL;DR

This paper explores new inequalities between key means of positive numbers using convexity properties of specific functions, leading to refined inequalities and applications to Ky Fan type inequalities.

Contribution

It introduces novel inequalities between important means based on convexity analysis and refines existing Ky Fan inequalities.

Findings

Derived new inequalities between means using convexity

Refined Ky Fan type inequalities with tighter bounds

Established relations applicable to positive real numbers

Abstract

In this paper, mainly using the convexity of the function $\frac{a^x-b^x}{c^x-d^x}$ and convexity or concavity of the function $\ln\frac{a^x-b^x}{c^x-d^x}$ on the real line, where $a>b\geq c>d>0$ are fixed real numbers, we obtain some important relations between various important means of these numbers. Also, we apply the obtained results to Ky Fan type inequalities and get some new refinements.