Abstract

The question of determining under which conditions the Schwarzian derivative of an algebroid function turns out to be a uniform meromorphic function in the plane is considered. In order to do this the behaviour of the Schwarzian derivative of an algebroid function $w(z)$ around a ramification point is analyzed. It is concluded that in case of a uniform Schwarzian derivative $S_{w}(z)$, this meromorphic function presents 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 Schwarzian derivative is presented and the question arises whether it contains all analytic algebroid functions with this property.

Keywords: Schwarzian derivative; algebroid function; Möbius transformation; ramification point.

MSC: 30B40, 30F99

Pages:  1--14     

Volume  74 ,  Issue  1 ,  2022