A sequence of sharp trigonometric inequalities
Volume 141 / 2015
Colloquium Mathematicum 141 (2015), 61-64
MSC: Primary 26D05.
DOI: 10.4064/cm141-1-6
Abstract
The purpose of this paper is to prove the following sequence of sharp trigonometric inequalities. Let $n\ge 5$ and $0< x< \pi$. Then \[\def\ffrac#1#2{#1/#2}\def\ff{\hskip1pt} \biggl(\cos\frac{x}{\sqrt{n-1}}\biggr)^{n-1} < \biggl(\frac{\sin\sqrt{\ffrac{5}{n}}\ff x}{\sqrt{\ffrac{5}{n}}\ff x}\biggr)^{\frac{3}{5}n} < \biggl(\cos\frac{x}{\sqrt{n}}\biggr)^n. \]
