An Alternative Proof of a Result of Bhayo and S\'{a}ndor

Abstract

In 2021, the second author of this paper pointed out that the method of applying some known results to prove Theorem $1.6$ in the paper, 'On certain old and new trigonometric and hyperbolic inequalities' by B. A. Bhayo and J. S\'{a}ndor contains a mistake and corrected this mistake by providing another simple proof. In this note, we aim to give an alternative simple proof by using series expansions of hyperbolic sine and hyperbolic cosine functions. Moreover, we propose sharp lower bounds for hyperbolic tangent function.