Let the function f(z) be regular or meromorphic in the domain D on the z-sphere. For the sake of definiteness we give first Definition 1.1. These problems are often closely connected with questions in conformal mapping and indeed in many cases have arisen from them.
Application to multivalent functions Bibliography 160 Authorlndex 168 Subject Index 169Ĭhapter One Introduction 1.1 The study of univalent functions to-day consists of the investigation of certain families of functions regular or meromorphic and univalent in prescribed domains which may be simply- or multiply- connected especially from the aspects of the values which they assume and extremal problems for their coefficients in power series expansions, function values and derivatives.
Relation to Dirichlet integrals and modules. New classes of problems: an example VIII. Families of Univalent Functions 122 Results on the inner radius for non-overlapping domains. Applications of the General Coefficient Theorem. Regions of values results for functions in S and their derivatives. Regions of values results for functions in Σ (D) and Σ, their derivatives and certain coefficients. 85 Proofs of the classical results and extensions. Parabolic, elliptic and hyperbolic slit mappings. Canonical Conformal Mappings 71 Circular, radial and spiral slit mappings. Discussion of the possibility of equality. Estimation of the area of its image from above and below. Statement of the General Coefficient Theorem. The General Coefficient Theorem 48 Definitions. Global structure of the trajectories on a finite oriented Riemann surface. Modules and Extremal Lengths 13 Fundamental definitions. His thanks are due also to Sister Barbara Ann Foos for the use of notes taken at the author's lectures in Geometric Function Theory at the University of Notre Dame in 1955 -1956. At the time of writing of this monograph the author has been receiving support from the National Science Foundation for which he wishes to express his gratitude. In a final chapter we give also a number of applications of the method of symmetrization. The central theme of our work is the General Coefficient Theorem which contains as special cases a great many of the known results on univalent functions. Also it should be said that there has been no attempt to provide an exhaustive bibliography, reference normally being confined to those sources actually quoted in the text. It should be mentioned also that the discussion of the method of the extremal metric is directed toward its application to univalent functions, there being no space to present its numerous other applications, particularly to questions of quasiconformal mapping. Nevertheless such is the power of the present method that it is possible to inclu de the great majority of known results on univalent functions. Apart from an introductory chapter in which a brief survey of the development of this theory is given there is therefore no attempt to follow up other methods of treatment. Preface This monograph deals with the application of the method of the extremal metric to the theory of univalent functions. Berlin Gottingen Heidelberg in 1958 ISBN 978-7-3 ISBN 978-7-7 (eBook) DOI 10.1007/978-7-7 BRUHLSCHE UNIVERSITATSDRUCKEREI GIESSEN JENKINS WITH 6 FIGURES SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1958Īlle Rechte, insbesondere das der Ubersetzung in fremde Sprachen, vorbehalten Ohne ausdriickliche Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (rhotokopie, Mikrokopie) zu vervielfaltigen © Springer-Verlag Berlin Heidelberg 1958 Originally published by Springer-Verlag, oHG.
UNIVALENT FUNCTIONS AND CONFORMAL MAPPING BY JAMES A. HEFT 18 = REIHE: MODERNE FUNKTIONENTHEORIE BESORGT VON L.V.AHLFORS SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1958.SPERNER = NEUE FOLGE - HEFT 18 = UNIVALENT FUNCTIONS AND CONFORMAL MAPPING BY JAMES A JENKINS WITH 6 FIGURES SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1958 Die Bezieher dee „Zentralbiatt fiir Mathematik" erhalten die „Ergebntoe der Mathematih" «u etnem gegenilber dem Ladenpreis um 10% ermufiigten VorzugapveUĮRGEBNISSE DER MATHEMATIK UND IHRERGRENZGEBIETE UNTER MITWIRKUNG DER SCHR1FTLEITUNG DES „ZENTRALBLATT FUR MATHEMATIK" HERAUSGEGEBEN VON L.V.DOOB-S.EILENBERG P.R. ERGEBNISSE DER MATHEMATIK UND IHRERGRENZGEBIETE UNTER MITWIRKUNG DER SCHRJFTLEITUNG DES „ZENTRALBLATT FUR MATHEMATIK" HERAUSGEGEBEN VON L.V.