On a functional mean inequality, application for the geometric operator mean

The geometric operator mean has been introduced by many authors in different ways. It has been defined as the unique positive solution of an algebraic Riccati equation, as a unique solution of an optimization problem as well as a strong limit of an operator sequence defined recursively by an iterative algorithm descending from the arithmetic and harmonic operator means. In this paper, we establish a functional mean inequality from which we derive another writing of the geometric operator mean in terms of the parameterized arithmetic and harmonic operator means.