Tight Exponential Bounds For Hyperbolic Tangent

Authors

  • Yogesh J. Bagul
  • Ramkrishna M. Dhaigude
  • Christophe Chesneau
  • Marko Kostić

Keywords:

Hyperbolic tangent, sigmoidal function, hyperbolic cosine, exponential bounds.

Abstract

In this article, we aim to obtain very tight exponential bounds for the hyperbolic tangent function. Our inequalities refine a double inequality recently proved by Zhang and Chen. In addition, graphical and numerical analysis are carried out, and a number of auxiliary lemmas may be of use on their own.

Key words and phrases. Hyperbolic tangent, sigmoidal function, hyperbolic cosine, exponential bounds.

2010 Mathematics Subject Classification. 26A09, 26D07, 33B10

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Published

2025-05-18

How to Cite

Yogesh J. Bagul, Ramkrishna M. Dhaigude, Christophe Chesneau, & Marko Kostić. (2025). Tight Exponential Bounds For Hyperbolic Tangent. Jordan Journal of Mathematics and Statistics, 15(4A), 807–821. Retrieved from https://jjms.yu.edu.jo/index.php/jjms/article/view/699

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