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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26 Pointwise domination. According to Littlewood's theorem, the subordination condition g "< f implies that f dominates g in the mean. We now turn to one of the deepest results in subordination theory, asserting that under an ' t appropriate normalization, If(z)l actually dominates Ig(z)b pointwise for all z in a certain subdisk. A similar statement can be made for derivatives. For these results it is essential that f be univalent. Theorem. 381 . . = This radius is best possible. This theorem has a long history.

3], [ 10]. ,f) (0 < 0 < 7r) . The fact that S'O >~ 0 is well known, and elementary. It is easy to check directly that T*(reiO,F) is harmonic in the upper half-plane. Conversely, we deduce from (20): Corollary. Let f(z) be meromorphic, of genus zero, with zeros {z n },poles {w n } and f(O) =# O,o°. I f T*(reiO,f) is harmonic, then f(z) = f(O)F(effXz) forsomereal cz and F(z) asin (19). The analogous statement still holds when f(z) has a zero or pole at the origin. This result can be used with (16) and a compactness argument to give an alternate proof of (11)-(12).

19). 17), we verify assertion (B) of the lemma. If we also use Rouch~'s lemma for meromorphic functions [5, p. 193], we obtain assertion (C). 18), denote by f(m) the ratio of ~"s and uo's which constitutes the central term of this double inequality. 20) t-2rr ,ogiao,+ ,og(pXm-°r(m)) = - -1 | 2rr J0 Pm(Pm eiO) log I Qm(Pm ei0) dO . 18). The proof of Lemma 2 is now complete. 2. Three normal families. 2) of the Pad~ polynomials. Take h ( 0 < 2 h < R ) so small that f(z) has no zeros and no poles in i z i ~ 2 h .

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