A proof of an open problem of Yusuke Nishizawa for a power-exponential function

Abstract

This paper presents a proof of the following conjecture, stated by Nishizawa in [Appl. Math. Comput. 269, (2015), 146--154.]: for $\displaystyle 0<x<\pi/2$ the inequality $ \displaystyle \frac{\sin{x}}{x} \!>\! \left(\frac{2}{\pi} + \frac{\pi\!-\!2}{\pi^{3}}(\pi^{2}\!-\!4x^{2})\right)^{\theta(x)}\! $ holds, where $\displaystyle \theta(x) \! = \! -\frac{(48\!-\!24\pi\!+\!\pi^{3})x^{3} }{3(\pi\!-\!2)\pi^{3}}+\frac{\pi^{3}}{24(\pi\!-\!2)}.$