most recent 30 from http://mathoverflow.net 2013-05-20T21:52:02Z http://mathoverflow.net/feeds/question/72821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72821/holomorphic-equation miriam 2011-08-13T09:10:43Z 2011-08-13T09:21:09Z

<p>hi, </p> <p>i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a smooth positive function. Does there exist any holomorphic function $f : U \rightarrow \mathbb{C}$ such that $f(z) \overline{f(z)} = c(z)$ ??? it would be very nice if somebody could help me.</p> <p>andrei</p>

http://mathoverflow.net/questions/72821/holomorphic-equation/72822#72822 Ben McKay 2011-08-13T09:21:09Z 2011-08-13T09:21:09Z

<p>Obviously $c$ would have to be real analytic. The Laplacian of $f(z)\overline{f(z)}$ must be positive, so $c$ would have to be a positive function with positive Laplacian. Clearly every positive power of the Laplacian on $c$ would also have to be a positive function. I don't know if that is sufficient.</p>