Sharp inequalities related to Wilker results

Source abstract

In this paper, we interested in Wilker inequalities. We provide finer bounds than known previous. Moreover, bounds are obtained for the following trigonometric function $$g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 - \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 0.$$