On a rigidity question related to spherical derivative of meromorphic functions 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 $f_1=\frac{pf_2-\bar{q}}{qf_2+\bar{p}}$.

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?
 A: The other possibility is that $f(z)=e^z$. Explicit computation gives
$$\frac{d}{dr}f^\#(re^{i\phi})=2e^{r\cos\phi}\cos\phi(1-e^{2r\cos\phi}),$$
which is $\leq 0$ everywhere.
On the other hand, the exponential is the only exception among entire functions.
Indeed, if $f^\#$ is decreasing on each radius, then $f^\#$ has no zeros,
this implies that $f'$ has no zeros. On the other hand, $f^\#(z)\leq f^\#(0)$ must be bounded, and a theorem of Clunie and Hayman implies that $f$ has at most exponential type (that is at most order 1, normal type). Then $f'$ has at most exponential type, and since it has no zeros, it must be $e^{az+b}$ by the Hadamard factorization theorem.
Refs. The original paper of Clunie and Hayman is
MR0192055
Clunie, J.; Hayman, W. K.
The spherical derivative of integral and meromorphic functions.
Comment. Math. Helv. 40 (1966), 117–148.
Their proof was much simplified (and generalized) in
MR2869124
Barrett, Matthew; Eremenko, Alexandre
Generalization of a theorem of Clunie and Hayman.
Proc. Amer. Math. Soc. 140 (2012), no. 4, 1397–1402.
Edit. One can describe all meromorphic functions with your property: there is probably nothing except $L(e^{az})$,
where $L$ is linear-fractional transformation.
Sketch of the proof for meromorphic functions. As before, we have
that $f$ has no critical points (that is $f'(z)\neq 0$ and all poles are simple), but now $f$ is of order at most 2, normal type. (By definition of the order of a meromorphic function in terms of Nevanlinna characteristic). Now we use the theorem of R. Nevanlinna which describes all meromorphic functions of finite order without
critical points. All such functions are ratios $f=w_1/w_2$ of two
linearly independent solutions of the differential equation
$$w''=Pw,$$
where $P$ is a polynomial. The order of the function is $(\deg P+2)/2$. The case $\deg P=0$ corresponds to an exponential, so it remains to consider orders $3/2$ (Airy functions) and $2$ (Weber functions).
Now there is an asymptotic theory
of these differential equations which gives a very precise asymptotic formula for solutions as $z\to\infty$. (In physics, this formula is called the WKB approximation). It implies that for orders $>1$, the spherical derivative of $f$ is unbounded. (This proof is independent of the previous proof for entire functions but uses deeper tools, like Nevanlinna theory).
Refs.
R. Nevanlinna, Uber Riemannsche Flachen mit endlich vielen Windungspunkten, Acta Math. 58 (1932) 295–373.
Expositions in English:
A. Eremenko, Entire and meromorphic solutions of ordinary differential equations, Chapter 6 in the book: Complex Analysis I, Encyclopaedia of Mathematical Sciences, vol. 85; Springer, NY, 1997, 141-153.
G. Gundersen, J. Heittokangas and A. Zemirni, Asymptotic integration theory for $f''+P(z)f=0$, Expositiones math., 40, 1 (2022) 94-126.
See also A. Eremenko, A Toda lattice in dimension 2 and Nevanlinna theory, J. Math. Phys., Anal. Geom., 31 (2007) 39-46 for a generalization of this theorem of Nevanlinna.
