New sharp Cusa--Huygens type inequalities for trigonometric and hyperbolic functions

Abstract

We prove that for $p\in (0,1]$, the double inequality% \begin{equation*} \tfrac{1}{3p^{2}}\cos px+1-\tfrac{1}{3p^{2}}<\frac{\sin x}{x}<\tfrac{1}{% 3q^{2}}\cos qx+1-\tfrac{1}{3q^{2}} \end{equation*}% holds for $x\in (0,\pi /2)$ if and only if $0<p\leq p_{0}\approx 0.77086$ and $\sqrt{15}/5=p_{1}\leq q\leq 1$. While its hyperbolic version holds for $% x>0$ if and only if $0<p\leq p_{1}=\sqrt{15}/5$ and $q\geq 1$. As applications, some more accurate estimates for certain mathematical constants are derived, and some new and sharp inequalities for Schwab-Borchardt mean\ and logarithmic means are established.