Optimal bounds for Toader mean in terms of general means

Journal of inequalities and applications

article

Qian Zhang¹  Bing Xu³  Maoan Han⁴ 

[1] Mathematics and Science College, Shanghai Normal University;School of Science, Southwest University of Science and Technology;Department of Mathematics, Sichuan University;College of Mathematics and Computer Science, Zhejiang Normal University

关键词: Toader mean;    Double inequality;    Optimal bounds;    Complete elliptic integral;   

DOI  :  10.1186/s13660-020-02384-y

学科分类：电力

来源: SpringerOpen

【 摘 要 】

In this paper, we present the best possible parameters $\alpha (r)$, $\beta (r)$ such that the double inequality $$\begin{aligned} {}[\alpha (r)M^{r}(a,b)+(1-\alpha (r))N^{r}(a,b)] ^{1/r} 0$ with $a\neq b$, where $$ \operatorname{TD}(a,b):= \int ^{\pi /2}_{0}\sqrt{a^{2}\cos ^{2}\theta +b^{2}\sin ^{2} \theta }\,d\theta $$ is the Toader mean, and M, N are means. As applications, we attain the optimal bounds for the Toader mean in terms of arithmetic, contraharmonic, centroidal and quadratic means, and then we provide some new bounds for the complete elliptic integral of the second kind.

【 授权许可】

CC BY