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}
}
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
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