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

TL;DR

This paper establishes new sharp inequalities involving trigonometric and hyperbolic functions, providing improved bounds and applications to mathematical constants and means, extending classical Cusa--Huygens inequalities.

Contribution

It introduces novel sharp inequalities for sine and cosine functions with precise parameter bounds, including hyperbolic versions, and applies these to refine constants and means.

Findings

Derived bounds for sine and cosine functions with specific parameter ranges

Established hyperbolic inequalities with sharp bounds

Improved estimates for mathematical constants and means

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.