Sharp Cusa type inequalities for trigonometric functions with two parameters

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.