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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