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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. 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