Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean
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 ] $p\in [2, 9/4]$ ,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-07-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-016-1113-1 |
| _version_ | 1817980406657449984 |
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| author | Zhen-Hang Yang Yu-Ming Chu Wen Zhang |
| author_facet | Zhen-Hang Yang Yu-Ming Chu Wen Zhang |
| author_sort | Zhen-Hang Yang |
| collection | DOAJ |
| description | 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 $\operatorname{TQ}(a,b)=(2/\pi)\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }d\theta$ . |
| first_indexed | 2024-04-13T22:53:32Z |
| format | Article |
| id | doaj.art-f140b5c408e0431bbbf862c7d44588c2 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-13T22:53:32Z |
| publishDate | 2016-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-f140b5c408e0431bbbf862c7d44588c22022-12-22T02:26:07ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-07-012016111010.1186/s13660-016-1113-1Monotonicity of the ratio for the complete elliptic integral and Stolarsky meanZhen-Hang Yang0Yu-Ming Chu1Wen Zhang2School of Mathematics and Computation Sciences, Hunan City UniversitySchool of Mathematics and Computation Sciences, Hunan City UniversityAlbert Einstein College of Medicine, Yeshiva UniversityAbstract 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 $\operatorname{TQ}(a,b)=(2/\pi)\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }d\theta$ .http://link.springer.com/article/10.1186/s13660-016-1113-1complete elliptic integralStolarsky meanToader meanToader-Qi mean |
| spellingShingle | Zhen-Hang Yang Yu-Ming Chu Wen Zhang Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean Journal of Inequalities and Applications complete elliptic integral Stolarsky mean Toader mean Toader-Qi mean |
| title | Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean |
| title_full | Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean |
| title_fullStr | Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean |
| title_full_unstemmed | Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean |
| title_short | Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean |
| title_sort | monotonicity of the ratio for the complete elliptic integral and stolarsky mean |
| topic | complete elliptic integral Stolarsky mean Toader mean Toader-Qi mean |
| url | http://link.springer.com/article/10.1186/s13660-016-1113-1 |
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