# Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean

@article{Yang2016MonotonicityOT, title={Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean}, author={Zhen-Hang Yang and Yu-Ming Chu and Wen Chu Zhang}, journal={Journal of Inequalities and Applications}, year={2016}, volume={2016}, pages={1-10} }

- Published 2016
DOI:10.1186/s13660-016-1113-1

In the article, we prove that the function r↦E(r)/S9/2−p,p(1,r′)$r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r')$ is strictly increasing on (0,1)$(0, 1)$ for p≤7/4$p\leq7/4$ and strictly decreasing on (0,1)$(0, 1)$ for p∈[2,9/4]$p\in [2, 9/4]$, where r′=1−r2$r'=\sqrt{1-r^{2}}$, E(r)=∫0π/21−r2sin2(t)dt$\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}(t)}\,dt$ is the complete elliptic integral of the second kind, and Sp,q(a,b)=[q(ap−bp)/(p(aq−bq))]1/(p−q)$S_{p, q}(a, b)=[q(a^{p}-b^{p})/(p(a^{q}-b… CONTINUE READING