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Journal of Function Spaces
Volume 2016, Article ID 5131907, 7 pages
http://dx.doi.org/10.1155/2016/5131907
Research Article

Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

1School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
2School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China

Received 13 September 2015; Accepted 11 January 2016

Academic Editor: Henryk Hudzik

Copyright © 2016 Ying-Qing Song 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.

Abstract

We present the best possible parameters and such that the double inequalities and hold for all with and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, , , , and are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of and , and .