TITLE: Finding all the holomorphic functions whose imaginary and real parts as a functions of $(x,y)$ satisfy the following?
QUESTION [0 upvotes]: The problem requires finding all holomorphic functions $f$ : $\Bbb C$ $\to$ $\Bbb C$, which satisfy : $$2u(x,y)v(x,y)\ge \frac {1}{1+x^2+y^2}$$
whereas $(x,y)\in \Bbb R^2$ and $u(x,y)=Re(f(x+iy))$ , $v(x,y)=Im(f(x+iy))$

I'm familiar with this kind of problems where the conditions holds an equality, which can be solved by applying Cauchy-Riemann Theorem, but here I'm not quite sure what to do.

REPLY [0 votes]: If $w= u+iv$ is entire then your assumed estimate implies that  $2uv = Im (w^2)$ is  bounded from below by $0 $.  Thus the entire function $f(z)=w^2(z)$ takes values exclusively in the upper half-plane $(Im(f) \geq 0$.  This is impossible for many reasons; one being  that $f(1/z)$ would violate the Casorati-Weierstrass Theorem. Another: $ |e^{i f(z)}| = e^{-Im(f)} \leq 1$ means that $e^{i f}$ is bounded and entire.
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