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 $0<x<\pi /2$ the inequality $\frac{\sin x}{x}>{\left(\frac{2}{\pi }+\frac{\pi -2}{{\pi }^3}({\pi }^2-4x^2)\right)}^{\theta (x)}$ holds, where $\theta (x)=-\frac{(48-24\pi +{\pi }^3)x^3}{3(\pi -2){\pi }^3}+\frac{{\pi }^3}{24(\pi -2)}.$

References in corpus (4)

• A Method of Proving a Class of Inequalities of Mixed Trigonometric Polynomial Functions
• Accurate approximations of some expressions involving trigonometric functions
• A proof of two conjectures of Chao-Ping Chen for inverse trigonometric functions
• New sharp Jordan type inequalities and their applications

Cited by in corpus (4)

• A new method for proving some inequalities related to several special functions
• One method for proving some classes of exponential analytic inequalities
• Refined estimates and generalizations of inequalities related to the arctangent function and Shafer's inequality
• Some improvements of Jordan-Steckin and Becker-Stark inequalities