ON ALGEBROID FUNCTIONS WITH UNIFORM SCHWARZIAN DERIVATIVE

Abstract

The question of determining under which conditions the Schwarzianderivative of an algebroid function turns out to be a uniform meromorphicfunction in the plane is considered. In order to do this the behaviour of theSchwarzian derivative of an algebroid function $w(z)$ around aramification point is analyzed. It is concluded that in case of a uniformSchwarzian derivative $S_w(z)$, this meromorphic functionpresents a pole of order two at the projection of the ramification point,with a rational coefficient ${\gamma }_{-2}$, where $0<{\gamma }_{-2}<1.$A class of analytic algebroid functions with uniform Schwarzianderivative is presented and the question arises whether it contains allanalytic algebroid functions with this property.