Sharp Cusa type inequalities for trigonometric functions with two parameters

TL;DR

This paper establishes new sharp inequalities of Cusa type for trigonometric functions involving two parameters, providing improved bounds and applications to inverse trigonometric functions and means.

Contribution

It determines optimal parameter ranges for Cusa type inequalities and introduces new sharp inequalities with applications to inverse trigonometric functions and means.

Findings

Derived best parameter conditions for inequalities

Established new sharp inequalities for sine and cosine functions

Applied results to inverse trigonometric functions and means

Abstract

Let $\left( p,q\right) \mapsto \beta \left( p,q\right) $ be a function defined on $\mathbb{R}^{2}$. We determine the best or better $p,q$ such that the inequality% \begin{equation*} \left( \frac{\sin x}{x}\right) ^{p}<\left( >\right) 1-\beta \left( p,q\right) +\beta \left( p,q\right) \cos ^{q}x \end{equation*}% holds for $x\in \left( 0,\pi /2\right) $, and obtain a lot of new and sharp Cusa type inequalities for trigonometric functions. As applications, some new Shafer-Fink type and Carlson type inequalities for arc sine and arc cosine functions, and new inequalities for trigonometric means are established.