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Tight Exponential Bounds For Hyperbolic Tangent

Yogesh J. Bagul (1), Ramkrishna M. Dhaigude (2), Christophe Chesneau *(3) and Marko Kostić (4)

(1) Department of Mathematics, K. K. M. College, Manwath, Dist: Parbhani-431505, India

Email address: yjbagul@gmail.com


(2) Department of Mathematics, Government Vidarbha Institute of Science and
Humanities, Amravati(M. S.)-444604, India

Email address: rmdhaigude@gmail.com


*(3) LMNO, University of Caen-Normandie, Caen, France

Email address: christophe.chesneau@unicaen.fr


(4) Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6,
21125 Novi Sad, Serbia

Email address: marco.s@verat.net

Doi : https://doi.org/10.47013/15.4.2

Cited by : Jordan J. Math & Stat., 15 (4A) (2022), 807 - 821


Received on: March 26, 2021;                                        Accepted on: March 8, 2022

 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.

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