# Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$? Answer

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## Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

let $$f(z)$$ be an entire holomorphic function in $$\mathbb{C}$$and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $$f(z)$$ is a polynomial, then it is easy to prove that $$\lim_{|z|\rightarrow \infty}g_f(z)=0$$.

When $$f$$ is transcendental, say $$f(z)=e^z$$then $$g_f(z)=\frac{e^x}{1+e^{2x}}$$which goes to zero when $$Re(z)$$ is going $$\infty$$but remains a constant when $$z$$ is moving on any verticle line. So far I have not found any example such that the limit goes to zero when $$f$$ is transcendental. My question is, can we rigorously prove that there is no transcendental entire function $$f$$ look for that $$\lim_{|z|\rightarrow \infty}g_f(z)=0$$?

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