On Lazarevic and Cusa type inequalities for hyperbolic functions with two parameters

TL;DR

This paper investigates inequalities involving hyperbolic functions with two parameters, generalizing known results and establishing new sharp inequalities, with applications to bivariate means.

Contribution

It introduces new inequalities for hyperbolic functions with two parameters, extending Lazarevic and Cusa type inequalities, and applies these to derive sharp inequalities for bivariate means.

Findings

Proved several theorems on inequalities for hyperbolic functions

Generalized known inequalities to broader parameter ranges

Presented new sharp inequalities for bivariate means

Abstract

In this paper, by investigating the monotonicity of a function composed of $% \left( \sinh x\right) /x$ and $\cosh x$ with two parameters in $x$ on $% \left( 0,\infty \right) $, we prove serval theorems related to inequalities for hyperbolic functions, which generalize known results and establish some new and sharp inequalities. As applications, some new and sharp inequalities for bivariate means are presented.