Abstract
Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $J(f) = J(a_{m_1}, a_{m_2},\dots, a_{m_n}) $ on some classes of univalent functions in the unit disk naturally connected with the canonical class $S$. The given functional $J$ is lifted to the Teichmuller space $\mathbf T_1$ of the
Results & Lemmas (6)
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Theorem 1.
Theorem 1. Any homogeneous polynomial functional (3), whose zero set ZJ is separated from the set (5), is maximized on bS(1) only by the…
Theorem 1. Any homogeneous polynomial functional (3), whose zero set ZJ is separated from the set (5), is maximized on bS(1) only by the functions f0 ∈K. In other words, any extremal function f0 of any homogeneous coefficient functional J on the class bS(1) is the Koebe function (4) composed with pre and post rotations about the origin, unless J(f0) = 0. The assumption ZJ ∩K = ∅cannot be omitted. The proof of this theorem involves a deep result from Teichm¨uller space theory given by the Bers isom
Corollary 1.
Corollary 1. If a functional J(f) on S satisfies max S |J(f)| = max bS(1) |J(f)| and |J(κ0)| > 0, then every extremal of this functional is…
Corollary 1. If a functional J(f) on S satisfies max S |J(f)| = max bS(1) |J(f)| and |J(κ0)| > 0, then every extremal of this functional is a a rotated Koebe function κθ with some θ ∈[−π, π). For a homogeneous J, any function κθ is extremal for J. The assumptions of Corollary 1 are satisfied, for example, by monomial functionals J(f) = aq1 m1 . . . aqn mn, qj ∈N, and by functionals of the form J(f) = P(an), where P(z) is a polynomial without free term
Corollary 1
Corollary 1, applied to J(f) = an, implies that for all f ∈S, max S |an| = |an(κθ)| = n. This is an alternate and direct proof of the…
Corollary 1, applied to J(f) = an, implies that for all f ∈S, max S |an| = |an(κθ)| = n. This is an alternate and direct proof of the Bieberbach conjecture. The second application concerns sharp estimation of the coefficients of Schwarzian deriva- tives Sf(z) = ∞ X 0 αnzn (|z| < 1) of on S. This problem also has been investigated by many authors.
Lemma 1.
Lemma 1. The function uθ(t) is subharmonic in the domain Dθ.
Lemma 1. The function uθ(t) is subharmonic in the domain Dθ.
Lemma 2.
Lemma 2. Let D be a finitely connected domain on the Riemann sphere bC. Assume that there are a set E0 of positive two-dimensional Lebesgue…
Lemma 2. Let D be a finitely connected domain on the Riemann sphere bC. Assume that there are a set E0 of positive two-dimensional Lebesgue measure and a finite number of points z1, z2, ..., zm distinguished in D. Let α1, α2, ..., αm be non-negative integers assigned to z1, z2, ..., zm, respectively, so that αj = 0 if zj ∈E0. Then, for a sufficiently small ε0 > 0 and ε ∈(0, ε0), and for any given collection of numbers wsj, s = 0, 1, ..., αj, j = 1, 2, ..., m, which satisfy the conditions w0j ∈D, |w0
Theorem 2.
Theorem 2. Let the polynomial P(z) have on the unit circle exactly one critical point z0. Then the correspomding functional (21) is…
Theorem 2. Let the polynomial P(z) have on the unit circle exactly one critical point z0. Then the correspomding functional (21) is estimated by max bS(1) |J(f)| = |J(κτ0,θ0)|, (22) where τ0 and θ0 are defined (in general, not uniquely) from the equations d|P(ei(ω+τ))| dω
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