If the inequality $f({\sqrt x}{\sqrt y})⩽{\sqrt f(x)}{\sqrt f(y)}$ is satisfied for all non negative $x,y$ in the domain, what can we say about the convexity of $f$ ? Or are there any other properties of this type of functions?
2026-03-27 08:41:47.1774600907
Jensen's inequality but with geometric mean
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If $f : \mathbb{R_+} \to \mathbb{R}_+$ such that $f$ is (weakly) decreasing,
$$f\left(\frac{x+y}{2}\right) \leq f(\sqrt{xy}) \leq \sqrt{f(x)f(y)} \leq \frac{f(x) + f(y)}{2}$$
implies that $f$ is (weakly) convex.
Without any restriction on the function $f$, you can't conclude anything. Take $f(x) = \sqrt{x}$ defined over $\mathbb{R}_+$. It satisfies $(xy)^{0.25} = f\left(\sqrt{xy}\right) \leq \sqrt{f(x)f(y)} = (xy)^{0.25}$ for all $x,y \in \mathbb{R}_+$ but is concave.