Criteria for Bounded Valence of Harmonic Mappings

Abstract

Abstract. In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S

Harmonic Schwarzian derivative and methods of approximation of zeros

“…Let us again argue as in the introduction to show an alternative way of getting F N H by involving the power series expansion of f . For a given zero α of f = h + g, let z be close enough to α so that, by using only the first terms in the power series of f , we can write (10)…”

On Becker's univalence criterion

“…The constant 1 is sharp, by the sharpness of Becker's univalence criterion. If one of these mentioned inequalities, with a slightly smaller right-hand-side constant, holds in an annulus r 0 < |z| < 1, then f is of finite valence [15]. Conversely to these univalence criteria, there exist absolute constants 0…”