If you consider $f=\frac{P}{Q}$ the quotient of two polynomial function (i.e. $P,Q\in \mathbb{C} [z]$) then $\frac{f′}{1+|f|^2}$ \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 $$\frac{f′}{1+|f|^2}=O\left(\frac{1}{z}\right)$$$\frac{f'}{1+|f|^2}=O\left(\frac{1}{z}\right)$$as z goes to infinity, is the quotient of two polynomial function? Of course considering this quotient come from the metric of the sphere, and my question could be is any parametrization of the sphere with such a decreasing is of finite type? 1 # holomorphic function with special decreasing property If you consider f=\frac{P}{Q} the quotient of two polynomial function (i.e. P,Q\in \mathbb{C} [z]) then \frac{f′}{1+|f|^2} decrease like \frac{1}{z}. My question is, is the converse true? is an meromorphic function(define on the whole plane) which satisfies$$\frac{f′}{1+|f|^2}=O\left(\frac{1}{z}\right)
as $z$ goes to infinity, is the quotient of two polynomial function?