The spherical derivative of a meromorphic function is defined as $$f^\#(z):=\frac{|2f'(z)|}{1+|f(z)|^2}.$$ The motivation is that given a piecewise smooth curve $\gamma$ in the complex plane, the length of $f\circ \gamma$ in the Riemann sphere is given by $\int_{\gamma} f^\#(z)d|z|$. When $f=az$ or $f=a/z$, where $a$ is a nonzero complex number, then $f^\#(z)$ is radially symmetric and is strictly decreasing along the radial direction.

I did know from literature (see for example Lehto's 1961 paper) that the behavior of $f^\#$ near an essential singularity of $f$ can reflect certain value distributional properties of $f$ near the singularity, but I'm trying to understand how the property of $f^\#$ can determine certain rigidity property of $f$.

For example, here is a standard question: if generally $f^\#(z)$ is strictly decreasing along radial directions, is it ture that $f$ must be of form $az$ of $a/z$ for some nonzero complex number $a$? Are there any other possibilities?