Growth of the gradient of $f(x+y) \leq f(x) f(y)$

by Severin Schraven   Last Updated July 12, 2019 10:20 AM - source

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}_{\geq 0}$ be a radial $C^2$ function which satisfies the following functional inequality

$$ f(x+y) \leq f(x) f(y) $$

Does there exist constants $c,d$ such that for all $x\in \mathbb{R}^3$ we have $$ \vert \nabla f (x) \vert \leq c e^{d\vert x\vert} $$ In other words

If $f$ solves $f(x+y) \leq f(x)f(y)$, does its gradient also grow at most exponentially? How about the Hessian?

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