A Problem of Bombieri on Univalent Functions

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Abstract

The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk that was finally proved by de Branges some 70 years later, drifted many complex analysts attention to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper by X. H. Dong that fell in our hands contained a way to tackle one of the problems of Bombieri on the behavior of

Results & Lemmas (4)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Theorem 1. Theorem 1. Dong [10] Let f(z) = z + P∞ n=2 anzn ∈S and set (6) 1 2 log f(z) z = ∞ X n=1 γnzn. Then for each n = 2, 3,... we have Mn(f) +…

Theorem 1. Dong [10] Let f(z) = z + P∞ n=2 anzn ∈S and set (6) 1 2 log f(z) z = ∞ X n=1 γnzn. Then for each n = 2, 3, . . . we have Mn(f) + Kn(f) ≤0 where (7)

Lemma 1. Lemma 1. (Lebedev-Milin Inequality [28]) Let the formal power series g(z) = ∞ X k=1 βkzk be given. Set exp g(z) = ∞ X k=0 pkzk.

Lemma 1. (Lebedev-Milin Inequality [28]) Let the formal power series g(z) = ∞ X k=1 βkzk be given. Set exp g(z) = ∞ X k=0 pkzk.

Lemma 2. Lemma 2. (Lemma 1, p.196 [18]) Let λ(t); t ≥0 be an arbitrary continuous real function except possibly for a ﬁnite number of…

Lemma 2. (Lemma 1, p.196 [18]) Let λ(t); t ≥0 be an arbitrary continuous real function except possibly for a ﬁnite number of discontinuities of the ﬁrst kind. Suppose that |λ(t)| ≤e−t; t ≥0. Then by setting Z ∞ 0 λ2(t)dt = (ν + 1 2)e−2ν; ν ≥0 we have

Lemma 3. Lemma 3. For f(z) = P∞ k=1 anzn ∈S we have (10) |a3 −a2 2 2 |2 + |a2|2 −5 = 4(|γ2|2 + |γ1|2 −5 4) ≤− √ 2(2 −|a2|)3/2.

Lemma 3. For f(z) = P∞ k=1 anzn ∈S we have (10) |a3 −a2 2 2 |2 + |a2|2 −5 = 4(|γ2|2 + |γ1|2 −5 4) ≤− √ 2(2 −|a2|)3/2.