A catalog record for this book is available from the British Library. Harmonic mappings in the plane / Peter Duren. Includes bibliographical references and index.

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula toâ€¦ (More)

The pseudohyperbolic metric is developed for the unit ball of Cn and is applied to a study of uniformly discrete sequences and Bergman spaces of holomorphic functions on the ball. Theâ€¦ (More)

We show that the zeros of the hypergeometric polynomials $$F\left( { - n,kn + 1;kn + 2;z} \right)$$ , $$k,n \in \mathbb{N}$$ , cluster on the loop of the lemniscate $$\left\{ {z:|z^k \left( {1 - z}â€¦ (More)

A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions.â€¦ (More)

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularityâ€¦ (More)

A simple proof is given for Nehariâ€™s theorem that an analytic function f which maps the unit disk onto a convex region has Schwarzian norm â€–S fâ€– â‰¤ 2. The inequality in sharper form leads to theâ€¦ (More)

In this note we study the zeros of solutions of differential equations of the form u + pu = 0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.â€¦ (More)

It is shown that an analytic function taking circles to ellipses must be a MÃ¶bius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic MÃ¶biusâ€¦ (More)

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to beâ€¦ (More)