Geometric convexity of an operator mean

Let $\sigma$ be an operator mean in the sense of Kubo and Ando. If the representation function $f$ of $\sigma$ satisfies $f_\sigma (t)^p\le f_\sigma(t^p) \text{ for all } p>1,$ then the operator mean is called a pmi mean. Our main interest is the class of pmi means (denoted by PMI). To study PMI, the operator mean $\sigma$, wherein $$f_\sigma(\sqrt{xy})\le \sqrt{f_\sigma (x)f_\sigma (y)}\quad (x,y>0)$$ is considered in this paper. The set of such means (denoted by GCV) includes certain significant examples and is contained in PMI. The main result presented in this paper is that GCV is a proper subset of PMI. In addition, we investigate certain operator-mean classes, which contain PMI.