# Proving a function is constant, if a sum of the real and imaginary parts is bounded

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?

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?