Some sharp Wilker type inequalities and their applications

TL;DR

This paper establishes sharp Wilker type inequalities involving sine and tangent functions and their hyperbolic counterparts, identifying precise conditions on parameters for the inequalities to hold or reverse, thereby unifying and extending previous results.

Contribution

The paper provides new sharp inequalities of Wilker type with exact parameter conditions, generalizing and unifying existing inequalities for trigonometric and hyperbolic functions.

Findings

Inequalities hold for specific p values depending on k.

Conditions for inequality reversal are precisely characterized.

Results unify previous inequalities and extend their scope.

Abstract

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,\pi /2) $ if and only if $p>0$ or $p\leq -% \frac{\ln (k+2) -\ln 2}{k(\ln \pi -\ln 2)}$. 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 $% p>0$ 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.