This monograph deals with the application of the method of the
extremal metric to the theory of 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. Nevertheless such is the power of the
present method that it is possible to include the great majority of
known results on univalent functions. 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. Also it should be said that
there has been no attempt to provide an exhaustive biblio graphy,
reference normally being confined to those sources actually quoted
in the text. 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. In a final chapter we
give also a number of appli cations of the method of
symmetrization. At the time of writing of this monograph the author
has been re ceiving support from the National Science Foundation
for which he wishes to express his gratitude. His thanks are due
also to Sister BARBARA ANN Foos for the use of notes taken at the
author's lectures in Geo metric Function Theory at the University
of Notre Dame in 1955-1956."
General
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