Complex Analysis: Proceedings of the Conference held at the by Lars V. Ahlfors (auth.), James D. Buckholtz, Teddy J.

By Lars V. Ahlfors (auth.), James D. Buckholtz, Teddy J. Suffridge (eds.)

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Additional info for Complex Analysis: Proceedings of the Conference held at the University of Kentucky, May 18–22, 1976

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We shall now prove that f o r most values o f t the terms e - g ( n ) + n t o f the series f o r F(e t) drop o f f rapidly f r o m the maximum term. 1) g'(v) = t . We shall call v the central index, although it need no longer be an integer. We apply Lemma 1 with h(x) = g'(x). 5) - - l(2x- (x -- w) dw 3v)'v~(v) > ~ v ( x - v)~(v). 7) e-g(n)+nt <~ exp (-1/2v(n - v ) ~ ( v ) } e-g(v)+vt (n > 2v, v ~ E) . 1), has a finite variation, by Lemma 1. 7) hold for all values of t which lie outside an exceptional set of finite measure.

Goluzin [ 15] offers the following construction. Let f(z) = z(1 + z) -2 and let g(z) = f(oo(z)), where o~(z)=k-l((1-~)k(z)), O If'(r) l for sufficiently small e > 0, since r 2 - 6r + 1 < 0 in this case. R eferences 1. A. Baernstein, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139-169. 2. M. Biernacki, Sur quelques majorantes de ~a th(~orie des fonctions univalentes, C.

London Math. Soc. 23 (1925), 481-519. 29 20. H. MacGregor, Majorization by univalent functions, Duke Math, J, 34 (1967), 95-102. 21. S. Robertson, A remark on the odd schlicht functions, Buff. Amer. Math. Soc. 42 (1936), 366-370. 22. S. Robertson, The generalized Bieberbach conjecture for subordinate functions, Michigan Math, J. 12 (1965), 421-429. 23. S. Robertson, Quasi-subordination and coefficient conjectures, Bull, Amer. Math. Soc. 76 (1970), 1-9. 24. S. Robertson, Quasi-subordinate functions, Mathematical Essays Dedicated to A.

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