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-β\left( p,q\right) +β\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.

29 pages

References in corpus (2)

• New sharp Jordan type inequalities and their applications
• Some new inequalities for trigonometric functions and corresponding ones for means