Georgian Mathematical Journal Volume 8 (2001), Number 3, 537–552

QUASICONFORMAL DEFORMATIONS OF HOLOMORPHIC FUNCTIONS

SAMUEL L. KRUSHKAL

Dedicated to the memory of N. I. Muskhelishvili on the occasion of his 110th birthday

Abstract. This paper deals with holomorphic functions from Bergman spaces B^{p} in the disk and provides the existence of deformations (variations) which do not increase the norm of functions and preserve some other prescribed properties. Admissible variations are constructed (for even integer p ≥ 2) using special quasiconformal maps of the complex plane (in line with a new approach to variational problems for holomorphic functions in Banach spaces).

2000 Mathematics Subject Classification: Primary: 30C35, 30C50, 30C62, 32A36. Secondary: 30C75. Key words and phrases: Quasiconformal map, Bergman space, Taylor coefficients, variation, inverse mapping theorem, extremal.

1. Introduction

Quasiconformal maps have now become one of the basic tools in many fields of mathematics and its applications. This paper promotes the development of a new approach to solving variational problems in Banach spaces of holomorphic functions, which recently has been outlined in [5], [6].

The methods adopted in those problems rely mainly on the integral representations of holomorphic functions by means of certain measures, which reduces the problems to the investigation of these measures. This method often encounters great difficulties. Our approach is completely different. It is based on the special kind of quasiconformal deformations of the extended complex plane ℂ̂ = ℂ ∪ {∞}, which ensure a controlled change of the norm of a given function under variation and also preserve some other prescribed properties of the function.

This approach was established in [5], [6] mainly for the Hardy spaces H^{p} in the disk with an even integer p and applied to some well-known conjectures concerning nonvanishing functions. The specific features of these spaces are used there in an essential way. The present paper deals with more general Bergman spaces B^{p} of holomorphic functions. Its purpose is to show the existence of deformations (variations) which do not increase the norm of a function and

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

preserve some other prescribed properties, for example, vary independent of a suitable number of the Taylor coefficients of this function. Such requirements are, of course, rather rigid.

Such problems naturally arise, for example, while studying nonvanishing holomorphic functions. These are closely connected with nonbranched holomorphic coverings.

To preserve nonvanishing, we construct quasiconformal deformations of the complex plane which fix the origin. This produces additional rigidity of variations, and we show how it can be resolved in the case of Bergman functions.

Recall that the Bergman space B^{p}, 1 < p < ∞, in the unit disk Δ = {z = x + iy ∈ ℂ : |z| < 1} consists of holomorphic functions f with the finite norm

∥f∥_{p} = (∫∫_{Δ} |f(z)|^{p}dxdy)^{1/p}. Let |x| denote the Euclidean norm in ℝ^{n}.

2. Main Existence Theorem

The next theorem gives a complete answer to the question on the existence of admissible deformations with controlled distortion of the norm and coefficients of the Bergman functions for even integers p.

Theorem 1. Let m and n be two fixed positive integers (m, n ≥ 1). For every

function f_{0}(z) = c_{0}^{0} + ∑_{k=j}^{∞} c_{k}^{0}z^{k} ∈ B^{2m} ∩ H^{∞} with c_{0}^{0} ≠ 0 and c_{j}^{0} ≠ 0 (1 ≤ j < n),

which is not a polynomial of degree n_{1} ≤ n, there exists a number ε_{0} > 0 such that for each ε ∈ (0, ε_{0}) and any points d = (d_{1}, d_{j+1}, …, d_{n}) ∈ ℂ^{n−j+1} and a ∈ ℝ satisfying the inequalities |d| ≤ ε, |a| ≤ ε, there is a quasiconformal automorphism h of ℂ̂, which is conformal at least in the disk

Δ′_{0} := {w : |w| < sup_{Δ} |f_{0}(z)| + 1}

and satisfies (a) h(0) = 0;

(b) h′(0) = 1 + d_{1};

rom(c) h^{(k+1)}(0) = k! d_{k}, k = j + 1, …, n;

(d) ∥(h ∘ F_{0})′(z)∥_{2m}^{2m} − ∥F′_{0}∥_{2m}^{2m} = ∥h′(F_{0})f_{0}∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = a,

where F_{0}(z) = ∫_{0}^{z} f_{0}(t)dt.

The map h can be chosen to have the Beltrami coefficient μ_{h} = ∂w̄h/∂_{w}h with ∥μ_{h}∥_{∞} ≤ M_{0}ε. The quantities ε_{0}, M_{0} as well as the bound of the remainder in (c) depend only on f_{0}, m and n.

Proof. We associate with the functions f(z) = ∑_{0}^{∞} c_{k}z^{k} ∈ B^{p} the complex Banach

space B̃^{p} of their primitive functions

F (z) = ∫_{0}^{z} f(t)dt = ∑_{0}^{∞} \frac{c_{k}}{k + 1}z^{k+1},  (1)

QUASICONFORMAL DEFORMATIONS OF HOLOMORPHIC FUNCTIONS 539

letting

∥F ∥_{p} := ∥F ∥B̃^{p} = ∥f∥_{B^{p}}.

Note that sup_{Δ} |F (z)| ≤ sup_{Δ} |f(z)|, so if f ∈ B^{p} ∩ H^{∞}, then F ∈ B̃^{p} ∩ H^{∞}. In the sequel, we use p = 2m.

Now define for the annulus

E = {w : R < |w| < R + 1},

with a fixed R ≥ sup_{Δ} |f_{0}(z)| + 1, the integral operators

T ρ = −\frac{1}{π} ∫∫_{E} \frac{ρ(ζ)dξdη}{ζ − w} , Πρ = ∂_{w}T = −\frac{1}{π} ∫∫_{E} \frac{ρ(ζ)dξdη}{(ζ − w)^{2}} ,

assuming ρ ∈ L_{q}(E), q ≥ 2, and regarding the second integral as a principal Cauchy value.

Let us seek the required quasiconformal automorphism h = h^{μ} of ℂ̂ in the form

h(w) = w − \frac{1}{π} ∫∫_{E} \frac{ρ(ζ)dξdη}{ζ − w} = w + T ρ(w),  (2)

with the Beltrami coefficient μ = μ_{h} supported in E, i.e., with ∥μ∥_{∞} < κ < 1 and μ(w) = 0 on ℂ\E. Substituting (2) into the Beltrami equation ∂w̄h = μ∂_{w}h (with ∂_{w} = (1/2) (∂_{u} − i∂_{v}), ∂w̄ = (1/2) (∂_{u} + i∂_{v}), w = u + iv), one obtains

ρ = μ + μΠμ + μΠ(μΠμ) + · · · .

As is well-known, this series is convergent in L_{q}(E) for some q > 2; thus the distributional derivatives ∂w̄h = ρ and ∂_{w}h = 1 + Πρ belong to L_{q}^{loc}(ℂ). This implies the estimates

∥ρ∥_{L_{q}(Δ_{r})} ≤ M_{1}(κ, r, q)∥μ∥_{L_{∞}(ℂ)}, ∥Πρ∥_{L_{q}(Δ_{r})} ≤ M_{1}(κ, r, q)∥μ∥_{L_{∞}(ℂ)};
∥h∥_{C(Δ_{r})} ≤ M_{1}(κ, r, q)∥μ∥_{∞}

in the disks Δ_{r} = {w ∈ ℂ : |w| < r} (r < ∞) and the smoothness of quasiconformal maps as the functions of parameters: if μ(z; t) is a C^{1}-smooth L_{∞}(ℂ)-function of a real (respectively complex) parameter t, then ∂_{w}h^{μ(·,t)} and ∂w̄h^{μ(·,t)} are smoothly ℝ-differentiable (respectively, ℂ-differentiable) L_{p} functions of t; hence, the map t ↦ h^{μ(·,t)}(z) is C^{1} smooth as an element of C(\overline{Δ}_{r}). For the proof of these results, which will be exploited below, see, e.g., [8], Ch.2; [1]; [4], Ch. 2.

Setting

⟨ν, φ⟩ = −π^{−1} ∫∫_{E} ν(ζ)φ(ζ)dξdη, (ν ∈ L_{∞}(E), φ ∈ L_{1}(E)),

one can rewrite (2) for |w| < R in the form

h(w) = w + T μ(w) + ω(w) = w + ∑_{k=0}^{∞}⟨μ, φ_{k}⟩w^{k} + ω(w)  (3)

with

φ_{k}(ζ) = 1/ζ^{k+1}, k = 0, 1, …; ∥ω∥_{C(Δ_{r})} ≤ M_{2}(κ, r)∥μ∥_{∞}^{2} (r < ∞).

Comparing (3) for w = F_{0}(z) with the above-given conditions (a), (b) and (c), one finds that the following relations are satisfied by the desired Beltrami coefficient μ:

\begin{aligned}⟨μ, φ_{0}⟩ + O(∥μ∥_{∞}^{2}) & = 0, \\ ⟨μ, φ_{1}⟩ + O(∥μ∥_{∞}^{2}) & = d_{1}, \\ ⟨μ, φ_{k+1}⟩ + O(∥μ∥_{∞}^{2}) & = d_{k}, k = j, …, n. \end{aligned}  (4)

On the other hand, the condition (d), together with (3), gives the equality

∥(h ∘ F_{0})′(z)∥_{2m}^{2m} = ∥(F_{0} + T ρ ∘ F_{0})′(z)∥_{2m}^{2m}

= ∫∫_{Δ} |f_{0}(z) + (T μ ∘ F_{0})′(z)|^{2m}dxdy + O(∥μ∥_{∞}^{2})

= ∫∫_{Δ} |f_{0}(z) − (Πμ ∘ F_{0})′(z)f_{0}(z)|^{2m}dxdy + O(∥μ∥_{∞}^{2})

= ∫∫_{Δ} [|f_{0}(z)|^{2} − 2 Re[\overline{f_{0}(z)}Πμ ∘ F_{0}(z)f_{0}(z)]

+ |Πμ ∘ F_{0}(z)|^{2}|f_{0}(z)|^{2}]^{m}dxdy + O(∥μ∥_{∞}^{2}) = ∥f_{0}∥_{2m}^{2m}

+ \frac{2m}{π} Re[∫∫_{E} μ(ζ)dξdη ∫∫_{Δ} \frac{|f_{0}(z)|^{2m+1}}{(ζ − F_{0}(z))^{2}} dxdy] + O_{m}(∥μ∥_{∞}^{2}).

Define

ϕ(ζ) = −2m ∫∫_{Δ} \frac{|f_{0}(z)|^{2m+1}}{(ζ − F_{0}(z))^{2}} dxdy,  (5)

then the previous equality takes the form

∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = Re⟨μ, ϕ⟩ + O_{m}(∥μ∥_{∞}^{2}).  (6)

The function ϕ is holomorphic in a domain D ⊂ ℂ̂ containing the disk Δ_{0}, and ϕ|E belongs to the span A_{2}^{0} of the system {φ_{k}}_{0}^{∞} in L_{2}(E). We also have

ϕ(ζ) ≢ 0

in D because for large |ζ| integral (5) expands to ϕ(ζ) = ∑_{k=2}^{∞} b_{k}ζ^{−k} with

b_{2} = 2mπ|c_{0}^{0}|^{2m+1} > 0.

We need a stronger fact.

Lemma 1. Under the assumptions of Theorem 1, the function ϕ does not reduce to a linear combination of the fractions φ_{1}, …, φ_{l}, l ≤ n, i.e., it cannot be of the form

ϕ(ζ) = ∑_{k=1}^{l} b_{k}ζ^{−k}, l ≤ n.  (7)

Proof. We modify the arguments exploited in [5]. They rely on the real analyticity of the L_{p}-norm.

Assuming the contrary that equality (7) holds, one obtains by (6) that any μ ∈ L_{∞}(E) with ∥μ∥ ≤ ε, ε → 0 and such that ⟨μ, ϕ_{k}⟩ = 0, k = 0, 1, …, l, satisfies the relation

∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = O(ε^{2}).  (8)

Fix an integer s > n such that c_{s}^{0} ≠ 0, and consider in L_{1}(E) the span A_{l,s} of the functions φ_{k}|E, with 0 ≤ k ≤ l and k > s. Then

inf{∥φ_{s} − χ∥_{L_{1}(E)} : χ ∈ A_{l,s}} = d > 0.

By the Hahn–Banach theorem, there exists μ_{0} ∈ L_{∞}(E) such that

⟨μ_{0}, χ⟩ = 0, χ ∈ A_{l,s}; ⟨μ_{0}, φ_{s}⟩ = 1, ∥μ_{0}∥_{∞} = d.

By (2) and (3), the quasiconformal homeomorphism h^{αμ_{0}} with |α| = ε now assumes the form

h^{αμ_{0}}(w) = w + αw^{s} + O(ε^{2}).  (9)

After replacing the element φ_{s}(w) by φ_{s,β}(w) = 1/(w − β)^{s} with small |β|, we obtain the system

φ_{k}(w), k = 0, 1, 2, …, k ≠ s; φ_{s,β}(ζ) = 1/(w − β)^{s},

which also constitutes a basis in A_{2}^{0}, and pass to the corresponding map

h_{β}^{αμ_{0}}(w) = w + α(w − β)^{s} + O(ε^{2})  (10)

with the same remainder as in (9). Then

h_{β}^{αμ_{0}} ∘ F_{0}(z) := ∑_{1}^{∞} C_{k}^{α,β}z^{k} = ∑_{1}^{s−2} \frac{c_{k−1}^{0}}{k} z^{k} + [\frac{c_{s−2}^{0}}{s − 1} − sαβ(c_{0}^{0})^{s−1} + ⋯]z^{s−1}

+ [\frac{c_{s−1}^{0}}{s} + α(c_{0}^{0})^{s} + ⋯]z^{s} + ⋯ + O(ε_{1}^{2}),

where

ε_{1} = \sqrt{|α|^{2} + |β|^{2}}.

Accordingly,

\begin{aligned}(h_{β}^{αμ_{0}} ∘ F_{0})′(z) & = ∑_{1}^{s−2} c_{k−1}^{0}z^{k−1} + [c_{s−2}^{0} − s(s − 1)αβ(c_{0}^{0})^{s−1} + ⋯]z^{s−2} \\ & + [c_{s−1}^{0} + sα(c_{0}^{0})^{s} + ⋯]z^{s−1} + ⋯ + O(ε_{1}^{2}). \end{aligned} (11)

In the disk Δ, let

f_{0}(z)^{m} = ∑_{1}^{∞} c_{k,m}^{0}z^{k}, [(h_{β}^{αμ_{0}} ∘ F_{0})′(z)]^{m} = ∑_{1}^{∞} c_{k,m}^{α,β}z^{k}.

Applying Parseval’s equality in L_{2}(Δ), one obtains

\begin{aligned}∥(h_{β}^{αμ_{0}} ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} & = ∥[(h_{β}^{αμ_{0}} ∘ F_{0})′]^{m}∥_{2}^{2} − ∥f_{0}^{m}∥_{2}^{2} \\ & = ∑_{k=0}^{∞} r_{k}^{2}(|c_{k,m}^{α,β}|^{2} − |c_{k,m}^{0}|^{2}) \end{aligned}  (12)

with r_{k}^{2} = π/(k + 1). The right-hand side of (12) is a nonconstant real analytic function of α and β. We have

c_{k,m}^{α,β} = c_{k,m}^{0} + αg_{k,m}(β) = c_{k,m}^{0} + O_{k}(α) + O_{k}(|αβ|),

where O_{k}(ε)/ε ≍ d with the bounds depending only on ∥f_{0}∥_{∞}. It follows from (11) that the linear term Cαβ in (12) cannot vanish identically for all small |αβ|. Therefore

∥(h^{αμ_{0}} ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = O(ε_{1}), O(ε_{1})/ε_{1} ≍ d,

which contradicts (8) for suitable choices of ε = |α| → 0 and β → 0.

Denote the scalar product in A_{2}^{0} by

⟨χ̄, φ⟩_{E} = ∫∫_{E} \overline{χ(z)}φ(z)dxdy.

By Lemma 1, the power series of ϕ in the disk Δ_{R}^{∗} must contain the powers ζ^{−k−1} with k > n. Thus the remainder

ψ(ζ) = ϕ(ζ) − ∑_{0}^{n} b_{k}ζ^{−k−1} = (∑_{0}^{j−1} + ∑_{s}^{∞})b_{k}ζ^{−k−1}, s ≥ n + 1,  (13)

does not vanish identically in Δ_{R}^{∗}. It also satisfies

⟨ψ̄, φ_{k}⟩_{E} = 0, k = j + 1, …, n.

Further, for any homeomorphism h = h^{μ} satisfying h(0) = 0, with μ ∈ L_{∞}(ℂ) vanishing on ℂ \ E, we have by (3) the expansion

h ∘ F_{0}(z) = (1 + ⟨μ, φ_{1}⟩)F_{0}(z) + ∑_{k=2}^{∞}⟨μ, φ_{k}⟩F_{0}(z)^{k} + O(∥μ∥_{∞}^{2});

hence,

(h ∘ F_{0})′(z) = (1 + ⟨μ, φ_{1}⟩)f_{0}(z) + ∑_{k=2}^{∞} k⟨μ, φ_{k}⟩F_{0}(z)^{k−1}f_{0}(z)

+ O(∥μ∥_{∞}^{2}) =: ∑_{k=1}^{∞} c_{k}^{∗}z^{k}, ∥μ∥_{∞} → 0.

This yields, in particular, that c_{0}^{∗} = c_{0}^{0}h′(0) and since

kF_{0}(z)^{k−1}f_{0}(z) = k(c_{0}^{0}z + \frac{c_{j}^{0}}{j + 1}z^{j+1} + ⋯)^{k−1}(c_{0}^{0} + c_{j}^{0}z^{j} + ⋯),

one concludes that

c_{j}^{∗} = (1 + ⟨μ, φ_{1}⟩)c_{1}^{0} + O(∥μ∥_{∞}^{2})

for j = 1, and

c_{j}^{∗} = (1 + ⟨μ, φ_{1}⟩)c_{j}^{0} + (j + 1)⟨μ, φ_{j+1}⟩(c_{0}^{0})^{j+1} + O(∥μ∥_{∞}^{2})

when j > 1, provided μ is chosen so that ⟨μ, φ_{k}⟩ = 0, k = 0, 1, …, j, and

⟨μ, φ_{j+1}⟩ = −π^{−1} ∫∫_{E} μ(ζ)ζ^{−j−2}dξdη ≠ 0.

Then

|c_{j}^{∗}|^{2} − |c_{j}^{0}|^{2} = 2(j + 1) Re{d_{j+1}c_{j}^{0}(c̄_{0}^{0})^{j+1}} + O(∥μ∥_{∞}^{2}).

Put here μ = tν with ∥μ∥_{∞} = 1 and t → 0 (t ∈ ℂ). Combining with (6) and (7), this results in

lim_{t→0} \frac{|c_{j}^{∗}|^{2} − |c_{j}^{0}|^{2}}{t} = 2(j + 1) Re\{c_{j}^{0}(c̄_{0}^{0})^{j+1}⟨ν, φ_{j+1}⟩\}
= 2(j + 1) Re{c_{j}^{0}(c̄_{0}^{0})^{j+1}b_{j+1}⟨ν, ψ⟩ ≠ 0,

which shows that

b_{j+1} ≠ 0.  (14)

We now choose the desired Beltrami coefficient μ of the form

μ = ξ_{0}φ̄_{0} + ξ_{1}φ̄_{1} + ∑_{k=j}^{n} ξ_{k}φ̄_{k+1} + τψ̄, μ|ℂ \ E = 0,  (15)

with the unknown constants

ξ_{0}, ξ_{1}, ξ_{j}, …, ξ_{n}, τ

to be determined from equalities (4) and (6). Substituting expression (15) into (4) and taking into account the mutual orthogonality of φ_{k} on E, one obtains for determining ξ_{k} and τ the nonlinear equations

k!d_{k} = ξ_{k}σ_{k+1}^{2} + O(∥μ∥_{∞}^{2}), k = 0, 1, j + 1, …, n,  (16)

with d_{0} = 0 and σ_{k+1} = ∥φ_{k+1}∥_{A_{2}^{0}} . A comparison of (15) and (6) gives the equation

∥(h ∘ F_{0})′(z)∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m}
= Re ⟨ξ_{0}φ̄_{0} + ξ_{1}φ̄_{1} + ∑_{k=j}^{n} ξ_{k}φ̄_{k+1} + τψ̄, ϕ⟩_{E} + O(∥μ∥_{∞}^{2}),  (17)

hence the only remaining equation is a relation for Re ξ_{j}, Im ξ_{j}, Re τ, Im τ. Thus we add to (17) three real equations to distinguish a unique solution of the above system. First of all we will seek ξ_{j} satisfying

ξ_{j}b_{j+1}σ_{j+1}^{2} = −ξ_{0}b_{0}σ_{0}^{2} − ξ_{1}b_{1}σ_{1}^{2} − ∑_{k=j+1}^{n} ξ_{k}b_{k+1}r_{k+1}^{2}  (18)

(which annihilates the main part of the increment of the magnitude ∑_{0}^{n} |c_{k}|^{2} after

deformation). Then (17) takes the form

∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = Re⟨τψ̄, ϕ⟩_{E} + O(∥μ∥^{2}),

and, letting τ be real, we obtain

∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = τκ + O(∥μ∥_{∞}^{2}).  (19)

Separating the real and imaginary parts in (16), (18) and adding (19), we obtain 2(n − j) + 5 real equalities, which define a nonlinear C^{1}-smooth (in fact, real analytic) map

y = W (x) = W′(0)x + O(|x|^{2})

of the points

x = (Re ξ_{0}, Im ξ_{0}, Re ξ_{1}, Im ξ_{1}, Re ξ_{j}, Im ξ_{j} + 1, …, Re ξ_{n}, Im ξ_{n}, τ)

in a small neighborhood U_{0} of the origin in ℝ^{2(n−j)+5}, taking the values

y =(Re d_{0}, Im d_{0}, Re d_{1}, Im d_{1}, Re d_{j+1}, Im d_{j+1}, …, Re d_{n}, Im d_{n},
∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m})

also near the origin of ℝ^{2(n−j)+5}. Its linearization

y = W′(0)x

defines a linear map ℝ^{2(n−j)+5} → ℝ^{2(n−j)+5} whose Jacobian differs from the product r_{0}^{2}r_{1}^{2} ∏_{j}^{n} r_{k+1}^{2} by a nonzero constant factor.

Therefore, x ↦ W′(0)x is a linear isomorphism of ℝ^{2(n−j)+5} onto itself, and one can apply to W the inverse mapping theorem. It implies the existence of a Beltrami coefficient μ of the form (15), for which we have the assertions of Theorem 1.

The relations (14) and (18) show a special role played by the first nonzero coefficient c_{j}^{0} of f_{0}, which cannot be replaced in general by some c_{k}^{0} with j < k < n.

3. Holomorphic Dependence on Coefficient Parameters d_{j+1}, …, d_{n}

3.1. As a consequence of the proof of Theorem 1 we obtain

Theorem 2. For a fixed small a ∈ ℝ, the Beltrami coefficient μ determined by (15) can be chosen to be a complex L_{∞}-holomorphic function of the parameters d_{1}, d_{j+1}, …, d_{n}.

Proof. This follows from the uniqueness of solutions of the system (16), (18), (19). Let |d| < ε and |a| < ε satisfy the assumptions of Theorem 1 and define the desired μ by (15). Fix a such a and a τ = τ_{0} > 0 found in the proof of Theorem 1 and solve for this

μ = ξ_{0}φ̄_{0} + ξ_{1}φ̄_{1} + ∑_{k=j}^{n} ξ_{k}φ̄_{k+1} + τ_{0}ψ̄

the complex system (16), (18) separately. We already know that it has a unique solution ξ_{0}, ξ_{j+1}, …, ξ_{n}, and by the inverse mapping theorem these ξ_{k}, hence μ, depend on the given d holomorphically.

In particular, the value d_{j} = h^{(j)}(0)/j! also moves holomorphically, simultaneously with the independent parameters d_{j+1}, …, d_{n}, and one can use by variations the openness of holomorphic maps.

3.2. If we need to construct a quasiconformal homeomorphism h satisfying

∥h′(F_{0})f_{0}∥_{2m}^{2m} = ∥f_{0}∥_{2m}^{2m},

i.e., for a = 0, then Lemma 1 again allows us to seek a Beltrami coefficient μ of the form (15) with unknown constants ξ_{0}, ξ_{j+1}, …, ξ_{n}, τ to be determined from equations (4) and

⟨μ, ϕ⟩ = 0.

In this case, the existence of a desired μ of such form is again ensured, for small ε, by the inverse mapping theorem, and moreover, this theorem provides also L_{∞} holomorphy of τ and μ in all parameters d_{j}, d_{j+1}, …, d_{n}, which move independently.

3.3. A modification of Theorem 1. Omitting the assertion (a) of Theorem 1 (i.e., that h(0) = 0), one can drop the requirement for the original function f_{0} to be bounded in Δ.

Indeed, for f ∈ B^{p} with p > 2, the Hölder inequality yields

∥f∥_{2} ≤ π^{(p−2)/p}∥f∥_{p};

and hence the image domain F (Δ), where f is the primitive function (1) for f, has a finite area (counting with multiplicity). Therefore this domain must have the exterior points.

Now, conjugating the desired quasiconformal homeomorphism h with a linear fractional transformation γ of ℂ̂, one reduces the proof to the case, where the primitive function F_{0}(z) = ∫_{0}^{z} f_{0}(t)dt is bounded in the unit disk, and this is, in fact, exactly what was used in the proof of Theorem 1. As a result, the homeomorphism h obtained is conformal in some domain containing F_{0}(Δ).

4. Infinitesimal Deformations

The following infinitesimal version of Theorem 1 admits a simpler proof.

Theorem 3. Given a function f_{0}(z) = ∑_{k=1}^{∞} c_{k}^{0}z^{k} ∈ B^{2m} ∩ H^{∞}, distinct from

a polynomial of degree n_{1} ≤ n, then for sufficiently small ε_{0} > 0 and ε ∈ (0, ε_{0}) we have that for every point d = (d_{2}, d_{3}, …, d_{n}) ∈ ℂ^{n−1} and a ∈ ℝ so that |d| ≤ ε, |a| ≤ ε, there is a quasiconformal homeomorphism h of ℂ̂, which is conformal in the disk D′_{0} and satisfies:

(a) h(0) = 0, h′(0) = 1;

(b) h^{(k)}(0) = k! d_{k} + O(ε^{2}), k = 2, 3, …, n;

(c) ∥h′(F_{0})f_{0}∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = a + O(ε^{2}), where again F_{0}(z) = ∫_{0}^{z} f_{0}(t)dt.

The Beltrami coefficient of h is estimated similar to Theorem 1.

Proof. By assumption, f_{0}(z) contains at least one nonzero term c_{s}^{0}z^{s} with s > n. Split the space A_{2}^{0} into the span

A_{l} = ⟨φ_{0}, φ_{1}, …, φ_{l}⟩

and its orthogonal complement

A_{l}^{⊥} = {g ∈ A_{2} : ⟨g, φ⟩_{E} = 0, φ ∈ A_{l}}

for 0 < l ≤ n and observe that A_{l}^{⊥} contains all convergent series g(ζ) = ∑_{k=s}^{∞} g_{k}ζ^{−k−1} in the disk Δ_{R}^{∗}, with g_{s} ≠ 0. Any such g determines the map

h_{ε,s}[g](w) := w − ε(T ḡ)^{(s−1)}(w) = w − \frac{ε(s − 1)!}{π} ∫∫_{E} \frac{\overline{g(ζ)}dξdη}{(ζ − w)^{s}}  (20)

if ε ∈ ℂ is chosen close to 0. Similarly to (3), the restriction of T ḡ onto the disk Δ_{R} = {|w| < R} assumes the form

T ḡ(w) = −\frac{1}{π} ∫∫_{E} \frac{\overline{g(ζ)}dξdη}{ζ − w} = −\frac{1}{π} ∑_{k=s}^{∞} ḡ_{k}σ_{k}^{2}w^{k}  (21)

and thus in (20)

(T ḡ)^{(s−1)}(w) = −\frac{1}{π} ∑_{k=s}^{∞} ḡ_{k}σ_{k}^{2} k(k − 1) ⋯ (k − s + 2) w^{k−s+1}.

It follows from (20) that

∥(h_{ε,s}[g] ∘ F_{0})′(z)∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = Re⟨ḡ, ϕ_{s}⟩_{E} + O(ε^{2}), ε → 0, (22)

with

ϕ_{s}(ζ) = −\frac{2ms!}{π} ∫∫_{Δ} \frac{|f_{0}(z)|^{2m+1}}{(ζ − F_{0}(z))^{s+1}} dxdy.  (23)

We first verify that as in Lemma 1, ϕ_{s} cannot be a linear combination only of φ_{0}, …, φ_{l} for some l ≤ n, i.e., the equality

ϕ_{s}(ζ) = ∑_{0}^{l} \frac{b_{k}}{ζ^{k+1}} , l ≤ n,  (24)

cannot take place. Assume that (24) holds, then the relation (22) must reduce to

∥(h_{ε,s}[g]′(F_{0})f_{0}∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = O(ε^{2}), ε → 0.  (25)

Define

I(f_{0}; g) = ∫∫_{Δ} |f_{0}(z)|^{2m}[(T ḡ)^{(s−1)} ∘ F_{0}]′(z)dxdy

= ∫∫_{Δ} |f_{0}(z)|^{2m+1}(T ḡ)^{(s)} ∘ F_{0}(z)dxdy.

Using (21), this integral is evaluated as follows:

I(f_{0}; g) = ∫_{0}^{1} ∫_{0}^{2π}|f_{0}(re^{iθ})|^{2m}[(T ḡ)^{(s−1)} ∘ F_{0}]′(re^{iθ})rdrdθ

= −\frac{1}{π} ∫_{0}^{1} rdr ∫_{|z|=r} f_{0}(z)^{m}\overline{f_{0}(\frac{r^{2}}{z̄} )^{m}} ×

× ∑_{k=s}^{∞} ḡ_{k}σ_{k}^{2} k(k − 1) ⋯ (k − s + 2)(k − s + 1)F_{0}(z)^{k−s}f_{0}(z)dθ

= −\frac{1}{πi} ∫_{0}^{1} rdr ∫_{|z|=r} (∑_{k=0}^{∞} c_{k}^{0}z^{k})^{m}(∑_{k=0}^{∞} \frac{c̄_{k}^{0}r^{2k}}{z^{k}} )^{m} ×

× ∑_{k=s}^{∞} ḡ_{k}σ_{k}^{2} k ⋯ (k − s + 1) (∑_{k=1}^{∞} \frac{c_{k−1}^{0}}{k} z^{k})^{k−s} ∑_{k=0}^{∞} c_{k}^{0}z^{k} \frac{dz}{z}

= ∫_{0}^{1}[−2ḡ_{s}σ_{s}^{2}s! (∑_{k=0}^{∞} |c_{k}^{0}|^{2}r^{2k})^{m} + ∑_{k=s+1}^{∞} ḡ_{k}B_{k}(r)]rdr.

This shows that by suitable choice of the numbers g_{s}, g_{s+1}, …, the corresponding function g ∈ A_{l}^{⊥} satisfies I(f_{0}; g) > 0, and then

\frac{∥h_{ε,s}[g] ∘ F_{0}∥_{2m}^{2m} − ∥F_{0}∥_{2m}^{2m}}{ε} = \frac{2 Re \{εI(F_{0}; g)\}}{ε} + O(ε) = O(1), ε → 0.

This shows that equalities (24) and (25) cannot occur for sufficiently small |ε| and thus the function

ψ_{s}(ζ) := ϕ_{s}(ζ) − ∑_{0}^{n} φ_{k}(ζ) = ∑_{k=n′}^{∞} b_{k}ζ^{−k−1}, n′ ≥ n + 1,

does not vanish identically in Δ_{R}^{∗}.

We now construct a linear functional L on the space A_{2} whose kernel is the space A_{n,s}^{⊥}, the orthogonal complement of

A_{n,s} = ⟨φ_{0}, φ_{1}, …, φ_{n}, ψ_{s}⟩.

Fix a complex number d_{1} with 0 < |d_{1}| < ε, and first define a linear functional L_{1} on A_{2} satisfying

L_{1}(φ_{0}) = 0, L_{1}(φ_{k}) = d_{k}, k = 1, 2, …, n,

with given d_{2}, …, d_{n} and

L_{1}(φ) = 0, φ ∈ A_{n}^{⊥} = ⟨φ_{0}, φ_{2}, …, φ_{n}⟩^{⊥}.

Noting that all φ ∈ A_{2} are of the form φ = ∑_{0}^{∞} η_{k}r_{k}^{−1}φ_{k} and by Parseval’s

equality ∥φ∥_{A_{2}}^{2} = ∑_{0}^{∞} |η_{k}|^{2}, the norm of L_{1} can be estimated (using also Schwarz’s

inequality) by

∥L_{1}∥ = sup_{∥φ∥_{A_{2}}=1} |L_{1}(φ)|

= sup_{∑_{0}^{∞} |η_{k}|^{2}=1} | ∑_{k=1}^{n} η_{k}d_{k}/r_{k}| ≤ sup_{∑_{0}^{∞} |η_{k}|^{2}=1} ∑_{1}^{n} |d_{k}|^{2}/r_{k}^{2} ∑_{0}^{n} |η_{k}^{2}| < M_{2}(n)ε.

There exists a function g_{1} ∈ A_{2} representing L_{1} so that L_{1}(φ) = ⟨ḡ_{1}, φ⟩_{E} and ∥g_{1}∥_{2} = ∥L_{1}∥ = O(ε), which defines a polynomial map h_{1} : D_{R} → ℂ by

h_{1}(w) = w − \frac{1}{π} ∫∫_{E} \frac{\overline{g_{1}(ζ)}dξdη}{ζ − w} = w − \frac{1}{π} ∑_{0}^{∞}⟨ḡ_{1}, φ_{k}⟩_{E}w^{k} = ∑_{1}^{n} d_{k}w^{k}.

Let

h_{1} ∘ F_{0}(z) = ∑_{1}^{∞} \frac{c_{k−1,1}}{k} z^{k},

and put

a_{1} = 2|c_{1}^{0}|^{2} Re d_{1} + ∑_{k=2}^{∞} \frac{|c_{k−1,1}|^{2} − |c_{k−1}^{0}|^{2}}{k^{2}} .  (26)

Now define another linear functional L_{2} on A_{2}, setting

L_{2}(ψ_{s}) = a − a_{1}

and L_{2}(φ) = 0 for φ⊥ψ_{s} and extending by the Hahn–Banach theorem from the space ⟨ψ_{s}⟩ ⨁⟨ψ_{s}⟩^{⊥}, where ⟨ψ_{s}⟩ denotes the span {tψ_{s} : t ∈ ℂ} of ψ_{s} onto A_{2}. Its norm is estimated by

∥L_{2}∥ ≤ M_{3}(1/δ)ε, δ = dist(ψ_{s}, ⟨ψ_{s}⟩^{⊥}) > 0.

The corresponding function g_{2} ∈ A_{2} for which

L_{2}(φ) = ⟨ḡ_{2}, φ⟩_{E}, ∥g_{2}∥_{2} = ∥L_{2}∥ = O(ε),

determines a holomorphic map D_{R} → ℂ by

h_{2}(w) = −\frac{(s − 1)!}{π} ∫∫_{E} \frac{\overline{g_{2}(ζ)}dξdη}{(ζ − w)^{s}}

= −\frac{1}{π} ∑_{k=s}^{∞}⟨ḡ_{2}, φ_{k}⟩_{E} k(k − 1) ⋯ (k − s + 2) w^{k−s+1}.

Define finally the map h = h_{1} + h_{2} : ℂ̂ → ℂ̂ corresponding to L = L_{1} + L_{2}. This map is holomorphic in the disk Δ_{R}, and it follows from above that

h(0) = 0, h′(0) = 1 + O(ε), h^{(k)}(0) = k!d_{k} + O(ε^{2}), k = 2, 3, …, n,

with given d_{k}, and

∥(h ∘ F_{0})′∥_{2m}^{2m} − ∥f_{0}∥_{2m}^{2m} = a + O(ε^{2}).

Restricting h to a smaller disk \overline{Δ}_{R_{1}} = \{|w| ≤ R_{1} < R\}, we get

∥h′(w) − 1∥_{\overline{Δ}_{R_{1}}} ≤ M_{4}(R_{1})ε, ∥h^{(j)}(w)∥_{\overline{Δ}_{R_{1}}} ≤ M_{4}(R_{1})ε, j = 2, 3, …, n.(27)

If ε is sufficiently small, h|\overline{Δ}_{R_{1}} is univalent in this disk and admits quasiconformal extensions across the circle {|w| = R_{1}} onto the whole complex plane. One can use, for example, its Ahlfors–Weill extension [2] with the harmonic Beltrami coefficient

μ_{h}(w) = −\frac{1}{2}\overline{S_{h}(\frac{R_{1}^{2}}{w} )} \frac{(|w|^{2} − R_{1}^{2})^{2}}{\overline{w}^{4}} , |w| > R_{1},

where

S_{h} = (h′′/h′)′ − (h′′/h′)^{2}/2

denotes the Schwarzian derivative of h on \overline{Δ}_{R_{1}}. It follows from (27) that ∥μ_{h}∥_{∞} = O(ε) as ε → 0.

The quasiconformal homeomorphism h constructed above satisfies all the assertions of the theorem except h′(0) = 1. To establish the latter property, observe that the parameter d_{1} has been chosen arbitrarily in the above proof, and take now those values of d_{1} for which equality (26) is reduced to

a_{1} = 0.

One obtains a linear equation for the ε-linear term of Re d_{1} by which this quantity is determined uniquely. On the other hand, the value of Im d′_{1} does not affect the linear part of the right-hand side of (26).

The corresponding homeomorphism h satisfies

h′(0) = 1 + O(ε^{2}),

and it remains to rescale h by passing to h(z)/h′(0). This completes the proof of Theorem 2.

5. Extremal Holomorphic Covering Maps

5.1. We present an application of the above theorems to a well-known coefficient problem for holomorphic functions.

Let F (z) = ∑_{1}^{∞} C_{k}z^{k} be locally injective in the unit disk. Then its derivative

F′(z) = f(z) does not vanish in Δ; thus F is a nonbranched holomorphic covering map Δ → F (Δ) normalized by F (0) = 0. Assume that F ∈ B̃^{2m} and consider the extremal problem to determine max |C_{n}| on the closed unit ball

B_{0}(B̃^{2m}) = {F (z) = ∑_{1}^{∞} C_{k}z^{k} ∈ B^{2m} : ∥F ∥ ≤ 1}.

It is equivalent to the Hummel–Scheinberg–Zalcman problem for nonvanishing Bergman holomorphic functions f in the disk (see [3]).

The first coefficient can be estimated in a standard way using Schwarz’s lemma for holomorphic maps Δ → B_{0}(B̃^{2m}). It gives, together with Parseval’s equality applied to f(z)^{m}, that |C_{1}| ≤ 1/\sqrt{π}. The equality takes place only for the function

F (z) = \frac{ϵz}{\sqrt{π}} , |ϵ| = 1.

However there is no such connection for higher coefficients.

Boundedness in the B̃^{p}-norm yields, by the mean value inequality, the compactness in the topology of locally uniform convergence on Δ, which, in turn, implies the existence of extremal covering maps F_{0} maximizing |C_{n}| on B_{0}(B̃^{2m}). Note that ∥F_{0}∥_{2m} = 1 since otherwise max |C_{n}| would increase by passing from F_{0} to (1 + r)F_{0} with appropriate r > 0.

Theorem 4. Any extremal map F_{0} maximizing |C_{n}| is of the form

F_{0}(z) = C_{1}^{0}z + ∑_{k=n}^{∞} C_{k}^{0}z^{k},

i.e., it satisfies C_{2}^{0} = ⋯ = C_{n−1}^{0} = 0 unless F_{0} is a polynomial of degree n (with nonvanishing derivative in Δ).

Proof. Let F_{0} be different from a polynomial of degree at most n. Then its Taylor series in Δ contains nonzero powers C_{s}^{0} with s > n. We assume the contrary that C_{j}^{0} ≠ 0 for some 1 < j < n and reach, by applying Theorem 1, a contradiction.

Observe that for any holomorphic function h in a domain D containing all the values F (z) for z ∈ Δ, we have h(w) = ∑_{1}^{∞} d_{k}w^{k} in a neighborhood of the

origin w = 0, and

h ∘ F_{0}(z) = d_{1} ∑_{1}^{∞} C_{k}^{0}z^{k} + d_{2}(∑_{1}^{∞} C_{k}^{0}z^{k})^{2} + ⋯ + d_{n}(∑_{0}^{∞} C_{k}^{0}z^{k})^{n} + ⋯

=: ∑_{1}^{∞} C_{k}^{∗}z^{k}.

Denoting F′_{0}(z) = f_{0}(z) = ∑_{0}^{∞} c_{k}^{0}z^{k} with c_{k}^{0} = (k + 1)C_{k+1}^{0}, we get

(h ∘ F_{0})′(z) = d_{1} ∑_{1}^{∞} kC_{k}^{0}z^{k−1} + 2d_{2} ∑_{1}^{∞} C_{k}^{0}z^{k} ∑_{1}^{∞} kC_{k}^{0}z^{k−1} + ⋯

+ nd_{n}(∑_{1}^{∞} C_{k}^{0}z^{k})^{n−1} ∑_{1}^{∞} kC_{k}^{0}z^{k−1} + ⋯

= d_{1} ∑_{1}^{∞} c_{k−1}^{0}z^{k−1} + 2d_{2} ∑_{1}^{∞} \frac{c_{k−1}^{0}}{k} z^{k} ∑_{1}^{∞} c_{k−1}^{0}z^{k−1} + ⋯

+ nd_{n}(∑_{1}^{∞} \frac{c_{k−1}^{0}}{k} z^{k})^{n−1} ∑_{1}^{∞} c_{k−1}^{0}z^{k−1} + ⋯

=: ∑_{0}^{∞} c_{k}^{∗}z^{k},

where

c_{0}^{∗} = d_{1}c_{0}^{0}, c_{1}^{∗} = d_{1}c_{1}^{0} + 2d_{2}(c_{0}^{0})^{2}, c_{2}^{∗} = d_{1}c_{2}^{0} + 3d_{2}c_{0}^{0}c_{1}^{0} + 3d_{3}(c_{0}^{0})^{3}, …. (28)

If h is, in addition, homeomorphic (thus conformal) on F_{0}(Δ), then the map h ∘ F_{0} is a local conformal homeomorphism of the disk Δ.

Now, let k = j_{0} > 1 and s ≥ n be the least indices of nonzero coefficients C_{k}^{0} for 1 < k < n and for k > n, respectively.

We apply Theorem 1 starting with j = j_{0} and a = 0, which provides a quasiconformal homeomorphism h of ℂ̂ satisfying the previous conformality assumptions and obtain from (28) that the coefficient C_{n}^{∗} becomes a holomorphic function of the independent parameters

d_{1}, d_{j_{0}+1}, …, d_{n+1}

defining h, because d_{j_{0}}, in itself, is a holomorphic function of these parameters. The openness of holomorphic maps allows us to choose these parameters (at least d_{1} and d_{n}) so that

|C_{n}^{∗}| = |d_{1}C_{n}^{0} + ⋯ + d_{n}(C_{0}^{0})^{n}| > |C_{n}^{0}| + O(ε) > |C_{n}^{0}|.  (29)

On the other hand, we have

∥(h ∘ F_{0})′∥_{2m} = ∥f_{0}∥_{2m} + O(ε^{2}) = 1 + O(ε^{2}),

hence the map h ∘ F_{0} + O(ε^{2}) also belongs to B_{0}(B̃^{2m}). But then (29) violates the extremality of F_{0} for C_{n} by a suitable choice of ε > 0. This proves C_{j_{0}}^{0} = 0.

Applying Theorem 1 successively for j_{0} = 2, 3, …, n − 1, one obtains the assertion of Theorem 4.

It seems likely that similar to what was established in [7] for the Hardy functions f ∈ H^{p}(Δ), the extremal map F_{0} in B_{0}(B̃^{2m}) cannot reduce to a polynomial of degree n_{1} ≤ n either.

5.2. The results of this paper extend to arbitrary Hilbert spaces of holomorphic functions in the disk whose norm is real analytic.

References

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2. L. V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations. Proc. Amer. Math. Soc. 13(1962), 975–978.

3. J. A. Hummel, S. Scheinberg, and L. Zalcman, A coefficient problem for bounded nonvanishing functions. J. Anal. Math. 31(1977), 169–190.

4. S. L. Krushkal, Quasiconformal mappings and Riemann surfaces. Wiley, New York, 1979.

5. S. L. Krushkal, Quasiconformal maps decreasing the L_{p} norm. Siberian Math. J. 41(2000), 884–888.

6. S. L. Krushkal, Quasiconformal homeomorphisms and Hardy functions. Preprint (2000).

7. T. J. Suffridge, Extremal problems for nonvanishing H^{p} functions. Computational Methods and Function Theory, Proceedings, Valparaiso 1989 (St. Ruscheweyh, E. B. Staff, L.C. Salinas, R. S. Varga, eds.), Lecture Notes in Math. 1435, 177–190, Springer, Berlin, 1990.

8. I. N. Vekua, Generalized analytic functions. (Russian) Fizmatgiz, Moscow, 1959; English translation: International Series in Pure and Applied Mathematics 25, Pergamon, New York, 1962.

(Received 7.05.2001)

Author’s address: Research Institute for Mathematical Sciences Department of Mathematics and Computer Science Bar-Ilan University, 52900 Ramat-Gan Israel E-mail: krushkal@macs.biu.ac.il