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International Journal of Mathematics and Mathematical Sciences
Volume 2014, Article ID 697643, 7 pages
Research Article

On the -Version of the Schwab-Borchardt Mean

Mathematical Research Institute, 144 Hawthorn Hollow, Carbondale, IL 62903, USA

Received 10 January 2014; Accepted 28 April 2014; Published 13 May 2014

Academic Editor: Kenneth S. Berenhaut

Copyright © 2014 Edward Neuman. 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.


This paper deals with a one-parameter generalization of the Schwab-Borchardt mean. The new mean is defined in terms of the inverse functions of the generalized trigonometric and generalized hyperbolic functions. The four new bivariate means are introduced as particular cases of the -version of the Schwab-Borchardt mean. For the particular value of the parameter , these means become either the classical logarithmic mean or the Seiffert means or the Neuman-Sándor mean. Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are also established.