Improvements of Inequalities for Sándor and Identric Means with Applications
1. Education Research Center, Education Bureau of Changxing County, Huzhou 313100, China
2. School of Civil Engineering, Huzhou Vocational and Technical College, Huzhou 313100, China
3. School of Data Science and Artificial Intelligence, Wenzhou University of Technology, Wenzhou 325000, China
| Abstract: | 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. |
| Keywords: | Sándor Mean; Identric Mean; Harmonic Mean; Arithmetic Mean; Inequality |
| DOI: | 10.57237/j.wjms.2025.02.002 |
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收稿日期: 2025-07-16
接受日期: 2025-09-29
在线出版日期: 2025-10-27
基金
| 1. | The Key Project of the Natural Science Foundation of the Department of Education of Zhejiang Province in 2020 (Grant no Y202043179) |
| 2. | The Zhejiang Province Teaching and Research Project in 2024 (Grant no 04588) |
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