A trigonometric double-inequality
Inequalities involving trigonometric functions have occupied attention of mathematicians for a long time. The purpose of this short article is to provide a certain generalization of an elegant double-inequality \[ |\sin (\cos x)| \leq |\cos x| \leq \cos (\sin x), \;\; \text{for every}\; x \in \mathbb R \] proved by \textit{R. J. Webster} that appeared in [Math. Gaz. 68, 37 (1984)]. The author of the paper under review shows that for all real numbers \(x\) one has \[ \alpha |\sin (\cos x) + \sin (\sin x)| \leq |\cos x + \sin x| \leq \beta (\cos (\cos x) + \cos (\sin x)), \] (\(\alpha , \beta\) are constants), with the best possible values of constant factors being \(\alpha = [\sqrt 2 \sin (1/\sqrt 2)]^{-1}\) and \(\beta = [\sqrt 2 \cos (1/\sqrt 2)]^{-1}.\)
scientific article; zbMATH DE number 5717766