# Journal of Operator Theory

Volume 57, Issue 2, Spring 2007 pp. 243-250.

The geometric means in Banach $*$-algebras**Authors**: Bao Qi Feng

**Author institution:**Department of Mathematical Sciences, Kent State University, Tuscarawas Campus, New Philadelphia, OH 44663, USA

**Summary:**The arithmetic-geometric-harmonic inequality has played a special role in elementary mathematics. During the past twenty five years (see [1], [2] and [8] etc.) a great many mathematicians have researched on various kinds of matrix versions of the arithmetic-geometric-harmonic inequality. It is interesting to see whether the arithmetic-geometric-harmonic inequality can be extended to the context of Banach $*$-algebras. In this article we will define the geometric means of positive elements in Banach $*$-algebras and prove that the arithmetic-geometric-harmonic inequality does hold in Banach $*$-algebras.

Contents Full-Text PDF