Some sharp inequalities involving Seiffert and other means and their concise proofs

TL;DR

This paper establishes new sharp inequalities involving various classical means of two positive numbers and provides concise proofs using monotonicity properties of functions related to sine and cosine.

Contribution

It introduces new sharp inequalities involving Seiffert and other means, with concise proofs based on monotonicity of trigonometric functions.

Findings

New sharp inequalities involving Seiffert and other means

Concise proofs using monotonicity of sine and cosine functions

Enhanced understanding of relationships among classical means

Abstract

In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert, contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square means of two positive real numbers $a$ and $b$ with $a\ne b$.