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On frame's inequalities.

Ling Zhu1

  • 1Department of Mathematics, Zhejiang Gongshang University, Hangzhou, China.

Journal of Inequalities and Applications
|May 1, 2018
PubMed
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This study corrects errors in analytic inequalities from a classic text. New inequalities for circular and hyperbolic functions are established, advancing mathematical understanding.

Area of Science:

  • Mathematical Analysis
  • Inequalities Theory

Background:

  • The foundational text "Analytic Inequalities" by Mitrinovic contains specific inequalities.
  • Theorem 3.4.20 in this text presents two key inequalities that require examination.

Purpose of the Study:

  • To identify and rectify errors within Theorem 3.4.20 of Mitrinovic's "Analytic Inequalities".
  • To reconstruct and establish corrected versions of these inequalities.
  • To extend the corrected inequalities to the domains of circular and hyperbolic functions.

Main Methods:

  • Rigorous mathematical analysis of the original inequalities.
  • Derivation and proof of corrected inequality statements.
  • Adaptation of the corrected inequalities using properties of trigonometric and hyperbolic functions.
Keywords:
Circular functionsCusa–Huygens inequalitiesFrame’s inequalitiesHyperbolic functions

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Main Results:

  • Specific errors in the two inequalities of Theorem 3.4.20 have been identified and corrected.
  • New, accurate inequalities for circular functions have been derived.
  • New, accurate inequalities for hyperbolic functions have been derived.

Conclusions:

  • The corrected inequalities provide a more precise mathematical framework.
  • The established circular and hyperbolic function inequalities offer valuable tools for further research in analysis.