International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 430692, 6 pages
http://dx.doi.org/10.1155/2012/430692
Research Article

A Nice Separation of Some Seiffert-Type Means by Power Means

Iulia Costin1 and Gheorghe Toader2

1Department of Computer Sciences, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
2Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania

Received 21 March 2012; Accepted 30 April 2012

Academic Editor: Edward Neuman

Copyright © 2012 Iulia Costin and Gheorghe Toader. 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

Seiffert has defined two well-known trigonometric means denoted by 𝒫 and 𝒯 . In a similar way it was defined by Carlson the logarithmic mean ℒ as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean ℳ . There are more known inequalities between the means 𝒫 , 𝒯 , and ℒ and some power means 𝒜 𝑝 . We add to these inequalities two new results obtaining the following nice chain of inequalities 𝒜 0 < ℒ < 𝒜 1 / 2 < 𝒫 < 𝒜 1 < ℳ < 𝒜 3 / 2 < 𝒯 < 𝒜 2 , where the power means are evenly spaced with respect to their order.