# 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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