Two question on harmonic function

Solution 1:

Let $f(z)$ be an entire function such that $\Re(f(z))=u(z)$.

First question. Let $g(z)=e^{f(z)}$. Then $$ 0<|g(z)|\le e^{a|\ln|z|\,|+b}=e^b|z|^a\qquad\forall z\in\mathbb{C}. $$ It follows that $g$ is a polynomial of degree $\le a$. Since it never vanishes, it must be a constant, and so must $f$ and $u$.

Second question We need to bound the modulus of a holomorphic function in terms of its real part. For this we use the Borel-Carathéodory theorem. From it it is easy to deduce that $$ |f(z)|\le C\,|z|^n $$ for some constant $C>0$. It follows that $f$ is a polynomial of degree $\le n$ and hence so is $u$.