Let $f:\mathbb R^n\to \mathbb R$ be a smooth function. Suppose that in a neighbourhood $U$ of $0$, we have two bounds: $$ |f(x)|\leq A $$ and $$ |\Delta f(x)|\leq B $$ Is it true that we have some bound of the form $$ |\nabla f(x)|\lesssim A^\theta B^{1-\theta}, $$ where $x\in U$?
The one-dimensional case is easier to verify. But I wonder if there is some high dimensional analogue.