Univalent Functions and Conformal Mapping

Abstract

Aile Re-chte, insbesondere das der Ubersetzung in fremde Sprachen, vorbehalten Oboe ausdruckliche Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (Photokople, Mikrokopie) zu vervielfaltlgen

Classification of complex polynomial vector fields in one complex variable

“…The second school and the one this paper belongs to is studying holomorphic vector fields in their own right (classification) in, for example, Brickman and Thomas [4] and Douady et al [8], and in the study of quadratic differentials in Jenkins [11] and Strebel [18]. Quadratic differentials and holomorphic vector fields can both be viewed as foliations with singularities.…”

“…Partial results for the global classification of complex polynomial vector fields go back to the classification of quadratic differentials having poles of order $ 2 [11,18]. The main case where classification of the global structure of quadratic differentials that applies to holomorphic vector fields is only a classification for given quadratic differentials.…”

Conformal mapping and ellipses

“…we refer to [2] for the definition and necessary properties of quadratic differentials. Let w = f (z) map a disc D on C conformally onto the interior of L. Let z 1 and z 2 be the preimages in D of w 1 and w 2 and let z 3 , z 4 be the reflections of z 1 , z 2 in the circle C = ∂D.…”

Univalent functions maximizing Re[f(ζ1) + f(ζ2)]

“…Let S denote the subset of H(A) that consists of functions that are univalent in A and satisfy f(0) 0 and if(O) 1. It is known [6] It is known ([4], [9], [7]) that each support point of S maps the disk onto the complement of a single analytic arc P with increasing modulus and an asymptotic direction at o. It was shown [11] that the omitted set F of a support point is a trajectory arc for the quadratic differential L(fe/(f-w))(dw/w)2, i.e.…”

“…It is known ( [7], [8]) that if 8 is an irrational multiple of 2 then every trajectory of (15) Ae '' is negative. Therefore e is negative.…”