{ "id": "1408.2243", "version": "v1", "published": "2014-08-10T15:51:12.000Z", "updated": "2014-08-10T15:51:12.000Z", "title": "New sharp Cusa--Huygens type inequalities for trigonometric and hyperbolic functions", "authors": [ "Zhen-Hang Yang" ], "comment": "15 pages", "categories": [ "math.CA" ], "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 $00$ if and only if $0