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Journal of Function Spaces
Volume 2016, Article ID 3698463, 6 pages
Research Article

Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

1Department of Mathematics and System Science, National University of Defense Technology, Changsha 410073, China
2Department of Mathematics, Changsha University of Science and Technology, Changsha 410014, China
3School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
4School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Received 8 May 2016; Revised 11 June 2016; Accepted 23 June 2016

Academic Editor: Rudolf L. Stens

Copyright © 2016 Hua Wang et al. 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.


We present the best possible parameters and such that the double inequalities , , hold for all with , where , , and are the arithmetic, quadratic, and Gauss arithmetic-geometric means of and , respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.