Advances in complex function theory: Proceedings of seminars by W. E. Kirwan, L. Zalcman

By W. E. Kirwan, L. Zalcman

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Extra resources for Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74

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3) n + s + i, of f i n i t e correct we = branch points of f on F 0' multiplicity. study as a n e t w o r k . the disection As v e r t i c e s of the we t a k e closed the plane finite by branch 28 points Vg, 9 = 1 multiplicity of order Let to s , q of together f on r0, which with ~, which we acts supposed as a b r a n c h to have point n - i. e be t h e total number of edges, to a v e r t e x , possibly the same 2e arcs r0 coming analytic branch point arcs. Summing of of order s of over all the one (namely out F0 each ~).

0, the I~I z [6], ~ = 1 to theorem Theorem < 1. the to equation v equation in the p l a n e f(z) the This that = w0 equa- proves and f(z) for a n y cannot p. F(~) Thus = t(~). have -v = w0 w 0. Now an e s s e n - is a p o l y n o m i a l . i, it r e m a i n s number To see this be the correspond set complement of k to s h o w that of t r a c t s we a r g u e v = 0. F0 of if Ivl as follows. Let in the k be the closed plane. 3) n + s + i, of f i n i t e correct we = branch points of f on F 0' multiplicity.

8 (1974), 279-282. 4. K. Hayman, The asymptotic behaviour of p-valent functions, Proc. London Math. Soe. 5 (1955), 257-284. 5. K. Hayman, Multivalent 1958. 6. J. Korevaar, Another numerical Tauberian theorem for power series, Nederl. Akad. Wetensch. Proc. Set. A. 57 -Indagationes Math. 16 (1954), 4 6 - 5 6 . 7. E. Landau, Darstellung und BeKr~ndung einiKer neuerer ErKebnisse der Funktionentheorie, Zweite Auflage, J. Springer, Berlin, 1929. 8. M. Milin, Haymants regularity theorem for the coefficients of univalent functions, Dokl.

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