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 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 $$
\lim_{|z|\rightarrow \infty}g_f(z)=0\; ?
$$
 A: This is not true. The optimal estimate from below for transcendental entire functions is
$$\limsup_{z\to\infty}\frac{|z||f'(z)|}{\log|z|(1+|f(z)|^2)}=\infty,$$
and this is best possible,
J. Clunie and W. Hayman, The spherical derivative of integral and meromorphic functons, Comment Math. Helv., 40 (1966) 117-148.
More precisely, for every function $\phi(r)\to+\infty$, they constructed an example of transcendental entire function
for which
$$\frac{|z||f'(z)|}{\phi(|z|)\log|z|(1+|f(z)|^2)}$$ is bounded. The function in this example is of the form
$$f(z)=\prod_{n=1}^\infty\left(1-\frac{z}{2^{k_n}}\right)^{k_n},$$
where $k_n$ is an increasing sequence of integers, which is choosen, depending on $\phi$.
A: A concrete example is given by $f(z)=\cos{\sqrt{z}}$. Then
$$
g_f(z) = \frac{|\sin\sqrt{z}|}{2|\sqrt{z}|(1+|\cos^2{\sqrt{z}}|)} .
$$
Obviously, this is small for large $|z|$ if $\sin{\sqrt{z}}$ is not large, and if $\sin\sqrt{z}$ is large, then $|\cos\sqrt{z}|$ is of the same order of magnitude (since $\sin^2 w+\cos^2 w=1$), and again $g_f$ is small.
