Assume $f$ is an entire function such that $$\lim_{z\to\infty}\frac{|f'(z)|}{1+|f(z)|^2}=0,$$ then should $f$ be a polynomial?
Picard's Theorem proves this instantly; which states:
Let $f$ be a transcendental (non-polynomial) entire function. Then $f-a$ must have infinitely many zeros for every $a$ (except for possibly one exception, called the lacunary value).
For from this theorem we see that there are infinitely many points (which can be chosen out of any given compact set, by virtue of isolated property of zeros) at which $|f|<1$, and at which $|f'|$ can be very large. But I want a more elementary proof for this.
This is a related question which I want to ask, and I think if the answer for my question is affirmative, then the linked question will be easily answered.