An Elementary Proof of the Caratheodory Kernel Convergence Theorem

M. Izuki, T. Koyama


Our aim is to give an elementary and self-contained proof of the Caratheodory kernel convergence theorem based on some fundamental facts in complex analysis.


the Caratheodory kernel convergence theorem, kernel of a sequence of domains, the Vitali theorem


L. Bieberbach, Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte

Abbildung des Einheitskreises vermitteln, S.-B. Preuss. Akad. Wiss. 38: 940{955,

L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154(1): 137{152,

J. B. Conway, Functions of One Complex Variable. II. Graduate Texts in Mathemat-

ics, 159. Springer-Verlag, New York, 1995.

P. L. Duren, Univalent Functions. Grundlehren der Mathematischen Wissenschaften

[Fundamental Principles of Mathematical Sciences], 259. Springer-Verlag, New York,

C. H. Fitzgerald and Ch. Pommerenke, The de Branges theorem on univalent func-

tions, Trans. Amer. Math. Soc. 290(2): 683{690, 1985.

H. Fujimoto, Complex Analysis, Iwanamishoten, 2006, Japanese.

W. K. Hayman, Multivalent Functions. Second edition. Cambridge Tracts in Mathe-

matics, 110. Cambridge University Press, Cambridge, 1994.

P. Henrici, Applied and Computational Complex Analysis Volume 3: Discrete Fourier

analysis{Cauchy integrals{construction of conformal maps{univalent functions. Pure

and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley

& Sons, Inc., New York, 1986.

Y. Komatsu, Function Theory, Asakurashoten, 1965, Japanese.

Ch. Pommerenke, Univalent Functions with a chapter on Quadraic Differentials by

Gerd Jensen. Studia Mathematica/Mathematische Lehrbucher, Band XXV. Vanden-

hoeck & Ruprecht, Gottingen, 1975.

S. L. Segal, Nine Introductions in Complex Analysis. Revised Edition. North-Holland

Mathematics Studies, 208. Elsevier Science B.V., Amsterdam, 2008.

Y. T^oki, Function Theory, Kyoritsushuppan, 1975, Japanese.

S. Gong, The Bieberbach Conjecture. Translated from the 1989 Chinese original and

revised by the author. With preface by Carl H. FitzGerald. AMS/IP Studies in Ad-

vanced Mathematics, 12. American Mathematical Society, Providence, RI; Interna-

tional Press, Cambridge, MA, 1999.

C. Caratheodory, Untersuchungen uber die konformen Abbildungen von festen und

veranderlichen Gebibeten, Math. Ann. 72(1): 107{144, 1912.

Y. Yoshida, Function Theory Second edition, Iwanamishoten, 1965, Japanese.

E. M. Stein and R. Shakarchi, Complex Analysis. Princeton Lectures in Analysis, 2.

Princeton University Press, Princeton, NJ, 2003.

L. V. Ahlfors, Complex Analysis. An introduction to the theory of analytic functions

of one complex variable. Third edition. International Series in Pure and Applied

Mathematics. McGraw-Hill Book Co., New York, 1978.

P. Henrici, Applied and Computational Complex Analysis. Volume 1: Power series integration-conformal mapping{location of zeros. Pure and Applied Mathematics.

Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

H. Sato, Complex Analysis, Kindaikagakusha, 1991, Japanese.

P. Henrici, Applied and Computational Complex Analysis. Volume 2: Special functions- integral transforms-asymptotics-continued fractions. Wiley Interscience [John Wiley & Sons], New York-London-Sydney, 1977.

M. Tsuji, Complex Function Theory, Makishoten, 1968, Japanese.


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