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 vertical 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 $f$ such that $$
\lim_{|z|\rightarrow \infty}g_f(z)=0\; ?
$$