In this paper, we provide alternative proofs to some results proposed in the article "New inequalities involving circular, inverse circular, hyperbolic, inverse hyperbolic and exponential functions" authored by Yogesh J. Bagul.

We obtain simple algebraic bounds of inverse hyperbolic sine and inverse hyperbolic tangent functions i.e., sinh⁻¹ x and tanh⁻¹ x. The inequalities are obtained on the entire domains of these functions. From our results, we obtain tighter bounds for the same functions. The Wilker and Huygens type inequalities involving inverse hyperbolic functions can also be easily derived from our main results.

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as sin−1xsinh−1x and tanh−1xtan−1x. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form eax2. Graph validation is also performed.

This paper deals with new inequalities involving the quotients (sin x)/(sinh x), (cos x)/(cosh x), and (tan x)/(tanh x).The proofs are based on l'Hôpital's rule of monotonicity, series expansions using Bernoulli numbers, and some analytical techniques. Some of the obtained inequalities have a resemblance with Adamović-Mitrinović, Wilker and Cusa-Huygens type inequalities.

Several bounds of trigonometric-exponential and hyperbolic-exponential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions.