Ji-A KIM and Toshiyuki SUGAWA

Geometric properties of functions with small Schwarzian derivative


The main aim in the present article is to give sufficient conditions for a locally univalent meromorphic function in the unit disk to have specific geometric properties such as starlikeness and convexity in terms of the Schwarzian derivative. To this end, we establish estimates of fundamental solutions to an ODE of the form 2y''+φ y=0 in the unit disk, where φ is an analytic function satisfying a given growth condition. As by-products, growth and distortion estimates are derived for a locally univalent strongly normalized analytic function f in the unit disk with a prescribed growth of the Schwarzian derivative.

submission: 23 February 2004

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