Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 349–357

EXACT COEFFICIENT ESTIMATES FOR UNIVALENT FUNCTIONS WITH QUASICONFORMAL EXTENSION

Samuel L. Krushkal

Bar-Ilan University, Research Institute for Mathematical Sciences Department of Mathematics, 52900 Ramat-Gan, Israel; krushkal@bimacs.cs.biu.ac.il

Abstract. We give here a complete solution of the coefficient problem for normalized univalent functions on the unit disk, with k -quasiconformal extension for a small k, and derive an explicit bound for k.

1. Introduction

While the coefficient problem is completely solved in the class of all normalized univalent functions on the disk [dB], the question remains open for functions with quasiconformal extension.

The strongest result here is established for the functions with k -quasiconfor-mal extension where k is small; see [Kr2].

Let S be the class of functions f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n} univalent in the unit disk Δ = {|z| < 1}. The class S(k) consists of f ∈ S admitting k -quasiconformal extension onto the whole Riemann sphere Ĉ = C ∪ {∞}, with additional normalization f̃(∞) = ∞. Let

\begin{aligned}f_{1}(z) & = \frac{z}{(1 − ktz)^{2}} ,  |z| < 1, |t| = 1, \\ f_{n−1} & = \{f_{1}(z^{n−1})\}^{1/(n−1)} = z + \frac{2kt}{n − 1}z^{n} + ⋯,  n = 3, 4, …. \end{aligned}

Consider on S a functional F of the form

F (f) = a_{n} + H(a_{m_{1}}, a_{m_{2}}, …, a_{m_{s}}),

where a_{j} = a_{j}(f); n, m_{j} ≧ 2 and H is a holomorphic function of s variables in an appropriate domain of C^{s}. We assume that this domain contains the origin 0 and that H, ∂H vanish at 0 .

The mentioned result of [Kr2] is:

1991 Mathematics Subject Classification: Primary 30C50, 30C75.

Theorem 1. For any functional of the above form there exists a k(F) > 0 such that, for k ≦ k(F),

(1)  max_{S(k)} |F (f)| = |F (f_{n−1})|

for some |t| = 1.

As a corollary, one immediately gets for f ∈ S(k) the sharp estimate

(2)  |a_{n}| ≦ \frac{2k}{n − 1}

for k ≦ k_{n}, with equality only for the function f_{n−1} . This solves the well-known problem of Kühnau and Niske; see [KuN]. The estimate (2) is interesting only for n ≧ 3, because for n = 2 there is the well-known bound |a_{2}| ≦ 2k for all k ∈ [0, 1] with equality for the function f_{1} .

The purpose of this paper is to improve on Theorem 1, supplementing it with an explicit estimate for the quantity k(F).

2. Statement of results

The main result of the paper is:

Theorem 2. Let sup_{S} |F (f)| = M_{n}. Then the equality (1) holds for all

(3)  k ≦ \frac{1}{2 + (n − 1)(M_{n} + 1)} =: k_{0}(F).

The bound (3) is not sharp and can be improved.

Corollary. The estimate (2) is valid for all

(4)  k ≦ \frac{1}{n^{2} + 1}.

Proof. Take F (f) = a_{n}. Since M_{n} = n, by de Branges’s theorem [dB], one immediately deduces from (3) that in this case

k_{0}(F) = \frac{1}{n^{2} + 1}.

For simplicity, we consider here the functionals F with holomorphic H depending on a finite number of coefficients a_{m}. The latter condition is not essential; one can take H depending on infinitely many a_{m} (provided the series expansion of H converges in some complex Banach space). The result shows that the main contribution here is given by the linear term a_{m}. The estimate (3) determines for which k this is true.

3. Proof of Theorem 2

We shall show that for k satisfying (3) all arguments employed in the proof of Theorem 1 in [Kr2] remain valid after some modification. Actually, we only need to modify the proof of Lemma 1.

On Δ^{∗} = {z ∈ Ĉ : |z| > 1} we have the Beltrami coefficients μ_{f} = ∂z̄f/∂_{z}f of the extensions f^{μ} of functions f ∈ S(k); these coefficients range over the ball

B(Δ^{∗}) = {μ ∈ L_{∞}(C) : μ | Δ = 0, ∥μ∥_{∞} < 1}.

Let B(Δ^{∗})_{k} = {μ ∈ B(Δ^{∗}) : ∥μ∥ ≦ k}.

Note that the Beltrami coefficient for f_{n−1} can be taken to be ktμ_{n}, where |t| = 1 and

(5)  μ_{n}(z) = \begin{matrix}|z| & n+1 \\  & z̄^{n+1} \end{matrix} .

We shall also use the following notations. For a functional L: S → C define

L̂(μ) = L(f^{μ}), μ ∈ B(Δ^{∗}).

If L is complex Gateaux differentiable, L̂ is a holomorphic functional on B(Δ^{∗}). All our functionals have this property.

For μ ∈ L_{∞}(Δ^{∗}), φ ∈ L_{1}(Δ^{∗}) we define

⟨μ, φ⟩ = −\frac{1}{π} ∫∫_{Δ^{∗}} μφ dx dy (z = x + iy).

For small k, the functions f^{μ} ∈ S(k) can be represented by

(6)  f^{μ}(ζ) = ζ − \frac{ζ^{2}}{π} ∫∫_{Δ^{∗}} \frac{μ(z) dx dy}{z^{2}(z − ζ)} + O(∥μ∥^{2}),

where the estimate of the remainder term is uniform on compact subsets of C (see e.g. [Kr1, Ch. 2]); this easily implies

F̂(μ) = ⟨μ, \frac{1}{z^{n+1}} ⟩ + O_{n}(∥μ∥^{2})

and hence

∥F̂′(0)∥ = sup\{|⟨μ, \frac{1}{z^{n+1}} ⟩| : ∥μ∥ ≦ 1\} = \frac{1}{π} ∫∫_{Δ^{∗}} \frac{dx dy}{|z|^{n+1}} = \frac{2}{n − 1}.

Now, applying the Schwarz lemma to the function

h_{μ}(t) = F̂(tμ) − F̂′(0)tμ: Δ → C,

where μ ∈ B(Δ^{∗}) is fixed, we get

(7)  |F̂(μ) − F̂′(0)μ| ≦ (M_{n} + ∥F̂′(0)∥)∥μ∥^{2} = (M_{n} + \frac{2}{n − 1})∥μ∥^{2}.

Consider the auxiliary functional

(8)  F̂_{p}(μ) = F̂(μ) + (p − 1)ξ⟨μ, \frac{1}{z^{p+1}} ⟩,

where p ≠ n is fixed and |ξ| < \frac{1}{2} . Then

(9)  sup_{B(Δ^{∗})} |F̂_{p}(μ)| < M_{n} + 1

and, similarly to (7),

(10)  |F̂_{p}(μ) − F̂′(0)μ − (p − 1)ξ⟨μ, \frac{1}{z^{p+1}} ⟩| ≦ (M_{n} + 1 + \frac{2}{n − 1})∥μ∥^{2}.

We shall require that

(11)  (M_{n} + 1 + \frac{2}{n − 1})∥μ∥^{2} < \frac{1}{n − 1}∥μ∥

or, equivalently,

(3′)  ∥μ∥ ≦ \frac{1}{2 + (n − 1)(M_{n} + 1)} = k_{0}(F).

Consider now any function f_{0} in S(k) maximizing |F | over S(k) (the existence of such functions follows from compactness). Let μ_{0} be an extremal dilatation of f_{0} , i.e.

∥μ_{0}∥_{∞} = inf{∥μ∥_{∞} ≦ k : f^{μ} | Δ = f_{0} | Δ}.

Note that ∥μ_{0}∥_{∞} = k by the maximum modulus principle. Suppose that μ_{0} ≠ ktμ_{n}, where |t| = 1, and μ_{n} is defined by (5). We show that this leads to contradiction for k satisfying (3). First of all, we may establish the following important property of extremal maps:

Lemma 1. If k satisfy (3), then for all 2 ≦ p ≠ n,

⟨μ_{0}, \frac{1}{z^{p+1}} ⟩ = 0.

Proof. Note that, from (6),

⟨μ_{0}, \frac{1}{z^{p+1}} ⟩ = lim_{τ→∞} \frac{a_{p}(f^{τμ_{0}})}{τ}.

Consider the classes S(τk_{0}) where k_{0} = k_{0}(F) is defined in (3) and 0 < τ < 1. It follows from (6) that, as τ → 0,

(12) max\{|F̂(μ)| : ∥μ∥ ≦ τk_{0}\} = \frac{τk_{0}}{π} ∫∫_{Δ^{∗}} \frac{dx dy}{|z|^{n+1}} + O_{n}(τ^{2}) = |F̂(τμ_{0})| + O_{n}(τ^{2}).

A similar calculation for functional (8) implies

(13)  max_{B(Δ^{∗})_{k}} |F̂_{p}(μ)| = \frac{τk_{0}}{π} ∫∫_{Δ^{∗}} |\frac{1}{z^{n+1}} + \frac{(p − 1)ξ}{z^{p+1}} | dx dy + O_{n}(τ^{2}),

where the remainder term estimate follows from (10) and depends (as in (12)) only on M_{n} and k_{0} .

Using the known properties of the norm

h_{p}(ξ) = ∫∫_{Δ^{∗}} |z^{−n−1} + (p − 1)ξz^{−p−1}| dx dy

following from the Royden [Ro] and Earle–Kra [EK] lemmas, we deduce from (12),

(13) that for small ξ there should be

(14)  max_{B(Δ^{∗})_{τk_{0}}} |F̂_{p}(μ)| = max_{B(Δ^{∗})_{τk_{0}}} |F̂(μ)| + τo_{p}(ξ) + O_{p}(τ^{2}ξ) + O_{n}(τ^{2}).

On the other hand, we have as ξ → 0, τ → 0, from (8)

\begin{aligned}|F̂_{p}(τμ_{0})| & = |F̂(τμ_{0})| + Re \frac{F̂(τμ_{0})}{|F̂(τμ_{0}|}(p − 1)ξ⟨τμ_{0}, \frac{1}{z^{p+1}} ⟩ + O(τ^{2}ξ^{2}) \\ & = |F̂(τμ_{0})| + τ(p − 1)|ξ||⟨μ_{0}, \frac{1}{z^{p+1}} ⟩| + O(τ^{2}ξ^{2}) \end{aligned}

with suitable choices of ξ → 0. Comparing this with (14), (10), (11), we conclude that ⟨μ_{0}, z^{−p−1}⟩ = 0. The proof of Lemma is completed.

This lemma is one of the central points in the proof of the Theorems 1 and 2. The crucial point in the proof of Lemma 1 is that we now have to check here that simultaneously an infinite (countable) number of ortogonality conditions remain valid for all k satisfying (3).

The next part of the proof is similar to [Kr2]. We briefly check that the arguments remain valid for all k.

Consider the Grunsky coefficients of the function \sqrt{f(z^{2})} which are defined from the series expansion

log \frac{(f(z^{2}))^{1/2} − (f(ζ^{2}))^{1/2}}{z − ζ} = − ∑_{m,n=1}^{∞} ω_{mn}z^{m}ζ^{n},

taking the branch of logarithm which vanishes at 1. The diagonal coefficients ω_{n−1,n−1}(f) are related to the Taylor coefficients of f by

(15)  ω_{n−1,n−1} = \frac{1}{2}a_{n} + P (a_{2}, …, a_{n−1})

where P is a polynomial without constant or linear terms (see [Hu]). Moreover, for f ∈ S(k) there is the well-known bound

|ω_{n−1,n−1}| ≦ \frac{k}{n − 1}

with equality only for the functions f_{n−1} .

Therefore, the map Λ_{n−1}: B(Δ^{∗}) → B(Δ^{∗}) defined by

Λ_{n−1}(μ) = {(n − 1)ω_{n−1,n−1}(μ)}μ_{n}

is holomorphic and fixes the disk {tμ_{n} : |t| < 1}. The differential of Λ_{n−1} at μ = 0 can be easily computed from (6), (15). It is an operator P_{n}: L_{∞}(Δ^{∗}) → L_{∞}(Δ^{∗}) given by

P_{n}(μ) = β_{n}⟨φ_{n}, μ⟩μ_{n}, φ_{n} = \frac{1}{z^{n+1}} .

Let us define P_{n}(μ) = α(k)μ_{n}. Since, by assumption, f_{0} is not equivalent to f_{n−1} , we have

\{Λ_{n−1}(\frac{t}{k} μ_{0}) : |t| < 1\} ⫋ \{| t |< 1\}.

Thus, by the Schwarz lemma,

(16)  |α(k)| < k.

Now consider the function

ν_{0} = μ_{0} − α(k)μ_{n}

and show that ν_{0} eliminates integrable holomorphic functions on Δ^{∗}. From Lemma 1 and the mutual orthogonality of the powers z^{m}, m ∈ z,

⟨ν_{0}, \frac{1}{z^{p+1}} ⟩ = 0

for p = 2, 3, …, p ≠ n. To establish that

⟨ν_{0}, \frac{1}{z^{n+1}} ⟩ = 0,

consider the conjugate operator

P_{n}^{∗}(φ) = β_{n}⟨μ_{n}, φ⟩φ_{n}, φ_{n} = \frac{1}{z^{n+1}} ,

which maps L_{1}(Δ^{∗}) onto L_{1}(Δ^{∗}) and fixes the subspace {λφ_{n} : λ ∈ C}. The definition of ν_{0} implies P_{n}(ν_{0}) = 0. Thus, for some λ,

⟨ν_{0}, φ_{n}⟩ = λ⟨ν_{0}, P_{n}^{∗}φ_{n}⟩ = λ⟨P_{n}ν_{0}, φ_{n}⟩ = 0.

Now consider in L_{1}(Δ^{∗}) the subspace A_{1}(Δ^{∗}) of functions φ which are holomorphic on Δ^{∗} and satisfy the condition φ(z) = O(|z|^{−3}) as |z| → ∞. Let

A_{1}(Δ^{∗})^{⊥} = {μ ∈ L_{∞}(Δ^{∗}) : ⟨μ, φ⟩ = 0 for all φ ∈ A_{1}(Δ^{∗})}.

Since the functions φ_{n} = 1/z^{n+1} , n = 2, 3, …, form a complete set in A_{1}(Δ^{∗}), we have proved that ν_{0} ∈ A_{1}(Δ^{∗})^{⊥}.

Now we use the well-known properties of extremal quasiconformal maps (see e.g. [Ga], [Kr1], [RS]). First of all, since μ_{0} is extremal for f_{0} ,

∥μ_{0}∥_{∞} = inf{|⟨μ_{0}, φ⟩| : φ ∈ A_{1}(Δ^{∗}), ∥φ∥ = 1};

moreover, such an equality is necessary and sufficient for μ ∈ B(Δ^{∗}) to be extremal for f^{μ}. Hence, for any ν ∈ A_{1}(Δ^{∗})^{⊥},

∥μ_{0}∥_{∞} = inf{|⟨μ_{0} + ν⟩| : φ ∈ A_{1}(Δ^{∗}), ∥φ∥ = 1} ≦ ∥μ_{0} + ν∥_{∞}.

Thus we have

Lemma 2. If f_{0} is extremal,

(17)  ∥μ_{0}∥_{∞} = k ≦ ∥μ_{0} − ν_{0}∥_{∞}.

We may now complete the proof of Theorem 2. By (17)

k ≦ ∥μ_{0} − ν_{0}∥_{∞} = ∥α(k)μ_{n}∥_{∞} = |α(k)|,

which contradicts (16). Hence f_{0} is equivalent to f_{n−1} and we can take μ_{0} = ktμ_{n} for some |t| = 1.

4. Complementary remarks and open questions

1) The estimates (1)–(3) also hold in the class S_{k}(1) of functions f ∈ S with k -quasiconformal extensions f̃ normalized by f̃(1) = 1.

The proof is similar, only (6) should be replaced with the corresponding representation formula for f ∈ S_{k}(1) [Kr1, Ch. 5]:

f^{μ}(ζ) = ζ − \frac{ζ^{2}(ζ − 1)}{π} ∫∫_{Δ^{∗}} \frac{μ(z) dx dy}{z^{2}(z − 1)(z − ζ)} + O(∥μ∥), as ∥μ∥ → 0.

2) Similar results are valid for the class Σ(k) of functions g(z) = z + ∑_{n=0}^{∞} b_{n}z^{−n}, z ∈ Δ^{∗}, with k -quasiconformal extensions to Ĉ which fix the origin.

The next two problems still remain open:

1) Does there exist an estimate of coefficients a_{n} (n ≧ 3) for f ∈ S(k) which holds for k ≦ k_{0} with a single k_{0} > 0?

2) Can we find exact estimates of coefficients a_{n} for univalent functions on the disk with quasiconformal extension in the general case when the dilatation k < 1 is arbitrary?

For f ∈ S(k), one gets from (7) the estimate

|a_{n}| ≦ \frac{2k}{n − 1} + (n + \frac{2}{n − 1})k^{2}

for any k, 0 ≦ k < 1, (cf. [KrKu, Part 1, Ch. 2]). Note also that Grinshpan [Gr] established the exact growth order, with respect to n, of the coefficients a_{n} of f ∈ S with k -quasiconformal extension, without any additional normalization: |a_{n}| ≦ cn^{k}.

References

[dB] de Branges, L.: A proof of the Bieberbach conjecure. - Acta Math. 154, 1985, 137–152.

[EK] Earle, C. J., and I. Kra: On sections of some holomorphic families of closed Riemann surfaces. - Acta Math. 137, 1976, 49–79.

[Ga] Gardiner, F. P.: Teichmüller theory and Quadratic Differentials. - John Wiley, New York, 1987.

[Gr] Grinshpan, A. Z.: On the growth of coefficients of univalent functions with quasiconformal extension. - Siberian Math. J. 23:2, 1982, 208–211.

[Hu] Hummel, J. A.: The Grunsky coefficients of a schlicht function. - Proc. Amer. Math. Soc. 15, 1964, 142–150.

[Kr1] Krushkal, S. L.: Quasiconformal Mappings and Riemann Surfaces. - V. H. Winston, Washington, 1979.

[Kr2] Krushkal, S. L.: The coefficient problem for univalent functions with quasiconformal extension. - In: Holomorphic Functions and Moduli I, Math. Sci. Research Inst. Publications 10, D. Drasin et al. (editors), Springer-Verlag, New York, 1988, 155–161.

Exact coefficient estimates 357

[KrKu] Kruschkal, S. L., und R. Kühnau: Quasikonforme Abbildungen - neue Methoden und Anwendungen. - Teubner-Texte zur Math. 54, Teubner, Leipzig, 1983.

[KuN] Kühnau, R., und W. Niske: Abschätzung des dritten Koeffizienten bei den quasikonform fortsetzbaren Funktionen der Klasse S. - Math. Nachr. 78, 1977, 185–192.

[RS] Reich, E., and K. Strebel: Extremal quasiconformal mappings with given boundary values. - In: Contributions to Analysis, L. V. Ahlfors et al. (editors), Academic Press, New York, 1974, 375–392.

[Ro] Royden, H. L.: Automorphisms and isometries of Teichmüller space. - In: Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud. 66, Princeton Univ. Press, Princeton, 1971, 369–383.

Received 17 March 1994