Some sharp Wilker type inequalities and their applications

Abstract

In this paper, we prove that for fixed $k\ge 1$, the Wilker type inequality {equation*} \frac{2}{k+2}(\frac{\sin x}{x}) ^{kp}+\frac{k}{k+2}(\frac{% \tan x}{x})^{p}>1 {equation*}% holds for $x\in (0,\pi /2)$ if and only if $p>0$ or $p\le -$. It is reversed if and only if $-\frac{12}{5(k+2)}\le p<0$. Its hyperbolic version holds for $x\in (0,\infty )$ if and only if or $p\le -\frac{12}{5(k+2)}$. And, for fixed $k<-2$, the hyperbolic version is reversed if and only if $p<0$ or $p\geq -\frac{12}{% 5(k+2)}$. Our results unify and generalize some known ones.

15 pages