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

TL;DR

This paper proves Nishizawa's conjecture that a specific inequality involving sine and a power-exponential function holds for all x in (0, π/2), confirming a long-standing open problem in mathematical analysis.

Contribution

The paper provides a rigorous proof of Nishizawa's conjecture, establishing the inequality for the first time and advancing understanding of sine-related inequalities with power-exponential functions.

Findings

Confirmed the inequality for all 0 < x < π/2

Validated the conjecture posed by Nishizawa in 2015

Contributed to the theory of inequalities involving trigonometric functions

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 $\displaystyle \theta(x) \! = \! -\frac{(48\!-\!24\pi\!+\!\pi^{3})x^{3} }{3(\pi\!-\!2)\pi^{3}}+\frac{\pi^{3}}{24(\pi\!-\!2)}.$