Extended a constant part of Redheffer's type inequalities

Article Sidebar

PDF
Published: Mar 25, 2018
DOI: https://doi.org/10.5556/j.tkjm.49.2018.2505
Keywords:
Redheffer's inequalities, monotonically increasing functions, monotonically decreasing functions, trigonometric functions

Main Article Content

Yusuke Nishizawa

Abstract

J-L. Li and Y-L. Li \cite{LL2007} gave the following Redheffer's type inequality; \begin{equation*} \frac{ 1 -\left( \frac{x}{\pi} \right)^2 }{ \sqrt{1 + 3 \left( \frac{x}{\pi} \right)^4}} > \frac{\sin{x}}{x} \end{equation*} holds for $0 < x < \pi$, where the constant $3$ is the best possible. In this paper, we establish two inequalities extended the constant part of the above inequality.

Article Details

How to Cite
Nishizawa, Y. (2018). Extended a constant part of Redheffer’s type inequalities. Tamkang Journal of Mathematics, 49(1), 79–83. https://doi.org/10.5556/j.tkjm.49.2018.2505
Issue
Vol. 49 No. 1 (2018)
Section
Papers
Author Biography

Yusuke Nishizawa

General Education, Ube National College of Technology, Tokiwadai 2-14-1, Ube, Yamaguchi 755-8555, Japan.

References

A. Baricz, Redheffer type inequality for Bessel functions,J. Inequal. Pure Appl. Math., 8(2007), no.1 Art.11, 6 pp.

C. P. Chen, J. W. Zhao and F. Qi, Three inequalities involving hyperbolically trigonometric functions, RGMIA Res. Rep. Coll., 6(2003), 437--443.

L. Li and J. Zhang, A new proof on Redheffer-Williams' inequality, 56(2011), 213-217.

J. L. Li and Y. L. Li, On the strengthened Jordan's inequality, J.Inequal. Appl., Art.ID 74328(2007), 8 pp.

J. L. Li, On a series of Erdos-Turan type, Analysis, 12(1992), 315--317.

K. Mehrez, Redheffer type inequalities for modified Bessel functions, Arab J. Math. Sci., 22(2016), 38--42.

R. Redheffer, P. Ungar, A. Lupas, et al., Problems and Solutions: Advanced Problems: 5642, 5665-5670, Amer. Math. Monthly, 76(1969), 422--423.

R. Redheffer and J. P. Williams, Solution of problem 5642, Amer. Math. Monthly, 76(1969), 1153--1154.

L. Zhu and J. Sun,Six new Redheffer-type inequalities for circular and hyperbolic functions, Comput. Math. Appl., 56(2008), 522--529.

L. Zhu, Sharpening Redheffer-type inequalities for circular functions,Appl. Math. Lett., 22(2009), 743--748.

L. Zhu, Extension of Redheffer type inequalities to modified Bessel functions, Appl. Math. Comput., 217(2011), 8504--8506.