Abstract and Applied Analysis
Volume 2012 (2012), Article ID 830585, 11 pages
http://dx.doi.org/10.1155/2012/830585
Research Article

Inequalities between Arithmetic-Geometric, Gini, and Toader Means

Yu-Ming Chu and Miao-Kun Wang

Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 24 August 2011; Accepted 20 October 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Yu-Ming Chu and Miao-Kun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We find the greatest values 𝑝 1 , 𝑝 2 and least values 𝑞 1 , 𝑞 2 such that the double inequalities 𝑆 𝑝 1 ( 𝑎 , 𝑏 ) < 𝑀 ( 𝑎 , 𝑏 ) < 𝑆 𝑞 1 ( 𝑎 , 𝑏 ) and 𝑆 𝑝 2 ( 𝑎 , 𝑏 ) < 𝑇 ( 𝑎 , 𝑏 ) < 𝑆 𝑞 2 ( 𝑎 , 𝑏 ) hold for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 and present some new bounds for the complete elliptic integrals. Here 𝑀 ( 𝑎 , 𝑏 ) , 𝑇 ( 𝑎 , 𝑏 ) , and 𝑆 𝑝 ( 𝑎 , 𝑏 ) are the arithmetic-geometric, Toader, and 𝑝 th Gini means of two positive numbers 𝑎 and 𝑏 , respectively.