In a question paper I got the following two questions.
- $u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex numbers. Show that $u$ is constant.
- If for all complex numbers, $u(z)\le |z|^n$ for some $n\in \mathbb{N}$, then $u$ is a polynomial in $x,y$.
I am completely stuck with this one.