Let $f$ be an entire function. Consider the set $$S=\bigg\{re^{i\theta}: r>0, \ \frac{\pi}{4}\leq \theta\leq \frac{7\pi}{4}\bigg\}\cup \{0\}.$$ It is given that $f$ is bounded on the set $S$. Suppose that Re$f(z)>0$ (real part of $f$ is positive) on the set $C\setminus S$. I want to show that any such function $f$ must be a constant with a positive real part.
My attempt: I know Liouville's theorem that every bounded entire function must be constant. The given function is bounded on the set $S$. Does this imply that $f$ is bounded everywhere?
I have picked many non-constant entire functions but could not find one that fulfills the given condition.
Please help which result can be applied here.
