New refinements of Becker-Stark inequality

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

$$\frac{{\pi }^2-({\pi }^2-8){\sin }^{2}x}{{\pi }^2-4x^2}<\frac{\tan x}{x}<\frac{{\pi }^2-(4-{\pi }^2/3){\sin }^{2}x}{{\pi }^2-4x^2}$$

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.