I was playing around with some functional equations and I came across the following question..
Suppose we have a $C^2(\mathbb R^2, \mathbb R)$ function $f$ with first and second derivatives globally bounded..
Moreover we know that $f(x, 0)=f(0, y)=0$ for any $x, y\in \mathbb R$.
Is it true then that there exists a constant $C> 0$ such that, for any $(x, y)\in\mathbb R^2$, we have
$|f(x, y)|\leq C|xy|$?