holomorphic functions and Laplace operator

Question

Prove that if $f,g$ are holomorphic in domain $D$ and doesn't have common zeros in it, then in this domain $∆\ln(|f|+|g|)\ge 0$. Where $∆f=\partial^2_x+\partial^2_y$. Is there any quick way, than just paint it?

Answer 1

If you know something about subharmonic functions: For constants $a$ and $b$ define $$u_{a,b}(z)=\ln(|af+bg|).$$ Then $u_{a,b}$ is subharmonic, being the log of the modulus of a holomorphic function (in fact on the set where $af+bg\ne0$ it is harmonic, since it's the real part of $\log(af+bg)$.)

And $u=\ln(|f|+|g|)=\sup_{|a|\le 1,|b\le 1}u_{a,b}$, so $u$ is subharmonic; now since $u$ is smooth this says $\Delta u\ge0$.