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

Abstract

Abstract In the article, we prove that the function r ↦ E ( r ) / S 9 / 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 − r 2 $r'=\sqrt{1-r^{2}}$ , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t $\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}(t)}\,dt$ is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 / ( p − q ) $S_{p, q}(a, b)=[q(a^{p}-b^{p})/(p(a^{q}-b^{q}))]^{1/(p-q)}$ is the Stolarsky mean of a and b. As applications, we present several new bounds for E ( r ) $\mathcal{E}(r)$ , the Toader mean T ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a 2 cos 2 t + b 2 sin 2 t d t ${T}(a,b)=(2/\pi)\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}\,dt$ , and the Toader-Qi mean TQ ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ $\operatorname{TQ}(a,b)=(2/\pi)\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }d\theta$ .

Keywords

• complete elliptic integral
• Stolarsky mean
• Toader mean
• Toader-Qi mean