Inequality regarding entire function

Question

Let $f$ be entire function. Must there exists $R>0$; such that $|f(z)| \leq |f'(z)|$ for all $|z|>R$ ,OR $|f'(z)| \leq |f(z)|$ for all $|z|>R$ ?

Answer 1

Hint: Pick a function such that both $f$ and $f'$ have infinitely many distinct zeroes...

$$f(z)=\sin(z)$$