New refinements of Becker-Stark inequality

Abstract

This paper deals with the well-known Becker-Stark inequality. By using variable replacement from the viewpoint of hypergeometric functions, we provide a new and general refinement of Becker-Stark inequality. As a particular case, the double inequality $ \begin{equation*} \frac{\pi^2-(\pi^2-8)\sin^2x}{\pi^2-4x^2}<\frac{\tan x}{x}<\frac{\pi^2-(4-\pi^2/3)\sin^2x}{\pi^2-4x^2} \end{equation*} $ for $ x\in(0, \pi/2) $ will be established. The importance of our result is not only to provide some refinements preserving the structure of Becker-Stark inequality but also that the method can be extended to the case of generalized trigonometric functions.

Keywords

• becker-stark inequality
• power series
• gaussian hypergeometric function
• monotonicity