Sharpening Redheffer-type inequalities for circular functions
type: Item
instance of: scholarly article
title: Sharpening Redheffer-type inequalities for circular functions
zbMATH Open document ID: 1172.26313
author: Ling Zhu
published in: Applied Mathematics Letters
publication date: 2009-06-09T00:00:00Z
review text: The author proves the following inequalities: (1) Let \(0<x\leq \pi\). Then \[ \Big(\frac{\pi^2-x^2}{\sqrt{\pi^4+3x^4}}\Big)^{\alpha} \leq \frac{\sin x}{x}\leq \Big(\frac{\pi^2-x^2}{\sqrt{\pi^4+3x^4}}\Big)^{\beta} \] holds if and only if \(\alpha \geq \pi^2/6\) and \(\beta \leq 1\). (2) Let \(0<x\leq \pi/2\). Then \[ \Big(\frac{\pi^2-4x^2}{\sqrt{\pi^4+48x^4}}\Big)^{\pi^2/6} \leq \cos x \leq \Big(\frac{\pi^2-4x^2}{\sqrt{\pi^4+48x^4}}\Big)^{3/4} \] holds. (3) Let \(0<x<\pi/2\). Then \[ \Big(\frac{\sqrt{\pi^4+48x^4}}{\pi^2-4x^2}\Big)^{1/2} \leq \frac{\tan x}{x} \leq \Big(\frac{\sqrt{\pi^4+48x^4}}{\pi^2-4x^2}\Big)^{\pi^2/6} \] holds.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 26D05
Mathematics Subject Classification ID: 33B10
zbMATH DE Number: 5563110
zbMATH Keywords: inequalities for trigonometric functions
zbMATH Keywords: Redheffer-type inequalities
MaRDI profile type: Publication
full work available at URL: j.aml.2008.08.012
OpenAlex ID: W2094055265
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DOI: 10.1016/J.AML.2008.08.012
description: scientific article; zbMATH DE number 5563110