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Liouville's theorem problem
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and suppose $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in\mathbb{C}$. How would you prove the function is constant?
I am approaching it by attempting to show it is bounded then by applying Liouville's Theorem. But have not made any notable results yet, any help would be greatly appreciated!