Optimal bounds for Toader mean in terms of general means

Journal of Inequalities and Applications

Maoan Han¹  Bing Xu²  Qian Zhang³ 

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

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

DOI  :  10.1186/s13660-020-02384-y

来源: DOAJ

【 摘 要 】

Abstract In this paper, we present the best possible parameters α ( r ) $\alpha (r)$ , β ( r ) $\beta (r)$ such that the double inequality [ α ( r ) M r ( a , b ) + ( 1 − α ( r ) ) N r ( a , b ) ] 1 / r < TD [ M ( a , b ) , N ( a , b ) ] < [ β ( r ) M r ( a , b ) + ( 1 − β ( r ) ) N r ( a , b ) ] 1 / r , $$\begin{aligned} {}[\alpha (r)M^{r}(a,b)+(1-\alpha (r))N^{r}(a,b)] ^{1/r} < &\operatorname{TD}\bigl[M(a,b),N(a,b)\bigr] \\ < &\bigl[\beta (r)M^{r}(a,b)+\bigl(1-\beta (r)\bigr)N^{r}(a,b) \bigr]^{1/r}, \end{aligned}$$ holds for all r ≤ 1 $r\leq 1$ and a , b > 0 $a,b>0$ with a ≠ b $a\neq b$ , where TD ( a , b ) : = ∫ 0 π / 2 a 2 cos 2 θ + b 2 sin 2 θ d θ $$ \operatorname{TD}(a,b):= \int ^{\pi /2}_{0}\sqrt{a^{2}\cos ^{2}\theta +b^{2}\sin ^{2} \theta }\,d\theta $$ 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】

Unknown