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Sharp inequalities for tangent function with applications.

Hui-Lin Lv1, Zhen-Hang Yang2, Tian-Qi Luo1

  • 1Department of Mathematics, Beijing Jiaotong University, Beijing, 100044 China.

Journal of Inequalities and Applications
|May 23, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces new bounds for a specific function, refining inequalities for the tangent function. These advancements provide sharper estimations for the Sine integral and Catalan constant.

Keywords:
Catalan constantSine integralinequalitiestrigonometric function

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Area of Science:

  • Mathematical Analysis
  • Inequalities
  • Special Functions

Background:

  • Existing Redheffer and Becker-Stark inequalities for the tangent function have limitations.
  • The need for refined bounds in analyzing special functions like the Sine integral and Catalan constant is recognized.

Purpose of the Study:

  • To establish novel, tighter bounds for a given function over a specified interval.
  • To derive improved estimations for the Sine integral and Catalan constant.
  • To introduce a new criterion for the monotonicity of power series quotients.

Main Methods:

  • Development of a new monotonicity criterion for the quotient of power series.
  • Application of this criterion to derive new function bounds.
  • Refinement of existing inequalities using the new analytical tools.

Main Results:

  • New bounds for the function [Formula: see text] on the interval [Formula: see text] were established.
  • Sharp estimations for the Sine integral and the Catalan constant were obtained.
  • The Redheffer and Becker-Stark type inequalities for the tangent function were refined.

Conclusions:

  • The newly developed monotonicity criterion is effective for refining inequalities.
  • The obtained bounds and estimations offer significant improvements over previous results.
  • This work contributes to the theory of inequalities and special function analysis.