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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

$|f(x, y)|\leq C|xy|$?

1 Answers 1

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Note

$f(x,y) = \int_0^x \int_0^y \frac{\partial^2 f}{\partial X\partial Y}\,dY\,dX.$

Thus if $|\partial^2 f / \partial X \partial Y| \leq C$ then $|f(x,y)|\leq C|xy|$.

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    If $|\partial f/\partial x| \leq C^\prime$ for all $x,y$ then $|f|\leq C^\prime |x|$, since $f(x,y) = \int_0^x \partial f/\partial X \,dX$. With *all* your hypotheses you can say there exists $C$ such that $|f|\leq C\min(|x|,|y|,|xy|)$.2012-08-20