Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant.
Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.