Improvements of Inequalities for Sándor and Identric Means with Applications

This paper is devoted to establishing the best possible upper and lower bounds for the Sándor mean and the identric mean in terms of the harmonic and arithmetic means. These two means, which are closely related to other classical means such as the logarithmic and Seiffert means, have attracted considerable attention in recent studies due to their rich mathematical structure and applications. The primary objective of this article is to refine existing inequalities by determining the sharpest constants in certain double inequalities involving convex combinations of the harmonic and arithmetic means and weighted geometric expressions. To achieve this, we apply a combination of classical inequality techniques and rigorous analysis of the monotonicity properties of auxiliary functions. Specifically, we construct appropriate comparison functions and examine their first and second derivatives to establish strict monotonicity and convexity. These analytic tools allow us to precisely identify the optimal parameters in the refined bounds for the Sándor and identric means. In particular, we show that the best possible bounds for these means can be expressed as weighted combinations of the expressions involving convex combinations of the harmonic and arithmetic means and weighted geometric expressions, where the weights are explicitly determined through monotonicity arguments. As an application of the main results, we derive new and sharp bounds for the functions of the inverse sine and the inverse hyperbolic tangent, which are important in various areas of mathematical analysis and approximation theory. Our findings improve upon several known results in the literature and provide a unified framework for analyzing the relationships among classical and non-classical means.