# An entire function satisfying $|f(z)|\le |z| |x|$ on $\mathbb{C}$ is identically zero.

Suppose $$f$$ is an entire function satisfying $$|f(z)|\le|z||x|$$ where $$x=\text{real part of }z$$. Then prove that $$f(z)=0$$ for every $$z\in \mathbb{C}$$

I am able to prove that $$f(0)=0$$ and $$f'(0)=0$$. I need to prove that all derivatives vanish at 0. How do I prove that? Help please.

The hypothesis implies that $$|f(z)| = 0$$ (and hence $$f(z) = 0$$) for all pure imaginary $$z$$. Since the imaginary axis is a set with a limit point, it follows by analytic continuation that $$f(z) = 0$$ for all $$z$$.