Some sharp Wilker type inequalities and their applications

In this paper, we prove that for fixed $k\geq 1$, 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 $x\in (0,π/2)$ if and only if $p>0$ or $p\leq -$. It is reversed if and only if $-\frac{12}{5(k+2)}\leq p<0$. Its hyperbolic version holds for $x\in (0,\infty)$ if and only if $$ or $p\leq -\frac{12}{5(k+2)}$. 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.

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