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 $f=az$, where $a$ is a nonzero complex number, then $f^\#(z)$ is radially symmetric and is strictly decreasing along the radial direction. Also, it is well known that $f_1^\#=f_2^\#$ is equivalent to the existence of complex numbers $p, q$ with $|p|^2+|q|^2=1$ such that $f_1=\frac{pf_2-\bar{q}}{qf_2+\bar{p}}$.

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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 a Mobius transform? Are there any other possibilities?