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Let $g$ be an entire function such that there exist three real numbers $a$, $b$, $c$ such that $a$ and $b$ are not both $0$ and such that $$a \operatorname{Re}(g(z)) + b \operatorname{Im}(g(z)) ≤ c$$ for all $ z ∈ \mathbb{C}$.

Show that $g$ is constant.

Solving simpler versions of this problem involves defining a new function, say $h(z)$, that is some (often exponential) function we can find to be bounded. What do we do in our case?

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Let $h(z)=(a-bi)g(z)$. Then$$(\forall z\in\mathbb{C}):\operatorname{Re}h(z)\leqslant c.$$Now, define $f(z)=e^{h(z)}$. Then$$(\forall z\in\mathbb{C}):\bigl\lvert f(z)\bigr\rvert=e^{\operatorname{Re}h(z)}\leqslant e^c.$$Can you take it from here?

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