Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means

Ji, Zhengchao ¹ ; Ding, Qing ² ; Zhao, Tiehong ³

¹ Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
² College of Mathematics and Statistics, Hunan University of Finance and Economics, Changsha 410205, China
³ Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 1, p. 150-160.

Résumé

In this paper, we present the best possible parameters $\alpha_i, \beta_i\ (i=1,2,3)$ and $\alpha_4,\beta_4\in(1/2,1)$ such that the double inequalities

$\alpha_1Q(a,b)+(1-\alpha_1)C(a,b) $

$\qquad\ Q^{\alpha_2}(a,b)C^{1-\alpha_2}(a,b) $

$\frac{Q(a,b)C(a,b)}{\alpha_3Q(a,b)+(1-\alpha_3)C(a,b)} $

$C\left(\sqrt{\alpha_4a^2+(1-\alpha_4)b^2},\sqrt{(1-\alpha_4)a^2+\alpha_4b^2}\right) $

hold for all $a, b>0$ with $a\neq b$, where $Q(a,b)$, $C(a,b)$, and $T(a,b)$ are the quadratic, contraharmonic, and Toader means, respectively, and $T_{Q,C}(a,b)=T[Q(a,b),C(a,b)]$. As consequences, we provide new bounds for the complete elliptic integral of the second kind.

Détail

DOI : 10.22436/jnsa.011.01.11

Classification : 26E60, 33E05
Keywords: Toader mean, elliptic integral, quadratic mean, contraharmonic mean

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