holomorphic function with special decreasing property If you consider $f=\frac{P}{Q}$ the quotient of two polynomial function (i.e. $P,Q\in \mathbb{C} [z]$) then $\frac{f'}{1+|f|^2}$ decrease like $\frac{1}{z}$. My question is, is the converse true? is an meromorphic function(define on the whole plane) which satisfies
$$\frac{f'}{1+|f|^2}=O\left(\frac{1}{z}\right)$$
as $z$ goes to infinity, is the quotient of two polynomial function?
Of course considering this quotient come from the metric of the sphere, and my question could be is   any parametrization of the sphere with such a decreasing is of finite type?
 A: The answer seems to be no.The quantity 
$$\rho(f(z)):= \frac{|f'(z)|}{1+|f(z)|^2}$$ is called the spherical derivative of $f$. Since you're interested in the behaviour of $z\rho(f(z))$ near $\infty$, then you should really take a look at Lehto and Virtanen's article :
MR0087747 (19,404a) Lehto, Olli; Virtanen, K. I. On the behaviour of meromorphic functions in the neighbourhood of an isolated singularity. Ann. Acad. Sci. Fenn. Ser. A. I. 1957 (1957), no. 240, 9 pp. (Reviewer: A. J. Lohwater), 30.0X
From the abstract :
It is proved first that if $f(z)$ is single-valued and meromorphic in a neighborhood of the isolated essential singularity $z=∞$, then there exists a universal constant $k>0$ such that $$\limsup_{z→∞}|z|\rho(f(z))≥k$$ for all such $f(z)$, while, for arbitrary $\epsilon>0$, there exist functions for which 
$$\limsup_{z \rightarrow \infty} |z|\rho(f(z)) < k+\epsilon.$$
A: Some one give an almost complete answer on mathstackechange, 
https://math.stackexchange.com/questions/73651/holomorphic-function-with-special-decreasing-property
the discussion should be continued there.
thanks all.
