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Convex and Starlike Mappings in Several Complex Variables (Paperback, Softcover reprint of the original 1st ed. 1998)
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Convex and Starlike Mappings in Several Complex Variables (Paperback, Softcover reprint of the original 1st ed. 1998)
Series: Mathematics and Its Applications, 435
Expected to ship within 10 - 15 working days
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This interesting book deals with the theory of convex and starlike
biholomorphic mappings in several complex variables. The underly-
ing theme is the extension to several complex variables of
geometric aspects of the classical theory of univalent functions.
Because the author's introduction provides an excellent overview of
the content of the book, I will not duplicate the effort here.
Rather, I will place the book into historical context. The theory
of univalent functions long has been an important part of the study
of holomorphic functions of one complex variable. The roots of the
subject go back to the famous Riemann Mapping Theorem which asserts
that a simply connected region n which is a proper subset of the
complex plane C is biholomorphically equivalent to the open unit
disk ~. That is, there is a univalent function (holo- morphic
bijection) I : ~ -+ n. In the early part of this century work began
to focus on the class S of normalized (f (0) = 0 and I' (0) = 1)
univalent functions defined on the unit disk. The restriction to
uni- valent functions defined on the unit disk is justified by the
Riemann Mapping Theorem. The subject contains many beautiful
results that were obtained by fundamental techniques developed by
many mathe- maticians, including Koebe, Bieberbach, Loewner,
Goluzin, Grunsky, and Schiffer. The best-known aspect of univalent
function theory is the so-called Bieberbach conjecture which was
proved by de Branges in 1984.
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