GROWTH AND COEFFICIENT ESTIMATES FOR UNIFORMLY LOCALLY UNIVALENT FUNCTIONS ON THE UNIT DISK

YONG CHAN KIM AND TOSHIYUKI SUGAWA

The present article is a summary of our paper [4] which will appear somewhere.

We will call a holomorphic function f on the unit disk 𝔻 uniformly locally univalent if f is univalent on each hyperbolic disk D(a, ρ) = \{z ∈ 𝔻; |\frac{z−a}{1−āz}| < tanh ρ\} with radius ρ and center a ∈ 𝔻 for a positive constant ρ. It is well-known (cf. [7]) that a holomorphic function f on the unit disk is uniformly locally univalent if and only if the pre-Schwarzian derivative (or nonlinearity) T_{f} = f′′/f′ of f is hyperbolically bounded, i.e., the norm

∥T_{f}∥ = sup_{z∈𝔻}(1 − |z|^{2})|T_{f}(z)|

is finite. This quantity can be regarded as the Bloch norm of the function log f′.

Because T_{f} is invariant under the post-composition by a non-constant linear function, we may assume that a holomorphic function f on the unit disk is normalized so that f(0) = 0 and f′(0) = 1. We denote by A the set of such normalized holomorphic functions on the unit disk. And we denote by B the set of normalized uniformly locally univalent functions: B = {f ∈ A; ∥T_{f}∥ < ∞}. The space B has a structure of non-separable complex Banach space under the Hornich operation ([6]). Also this space is important in connection with the Teichmüller theory (cf. [1] and [9]). The amount of the norm ∥T_{f}∥ is thought to be strongly reflected by some geometric or analytic properties of the function f, we will concern this quantity in the following.

For a non-negative real number λ we set

B(λ) = {f ∈ A; ∥T_{f}∥ ≤ 2λ},

here the number 2 is due to some technical reason.

In the class B(λ) for 0 ≤ λ < ∞ the function

F_{λ}(z) = ∫_{0}^{z} (\frac{1 + t}{1 − t})^{λ} dt

is extremal as we shall see later. We note that F_{λ} is univalent if and only if 0 ≤ λ ≤ 1. The following elementary fact is important for our argument below.

Theorem 1 (Distortion Theorem). Let λ be a non-negative real number. For an f ∈ B(λ) it holds that

F′_{λ}(−|z|) = (\frac{1 − |z|}{1 + |z|})^{λ} ≤ |f′(z)| ≤ (\frac{1 + |z|}{1 − |z|})^{λ} = F′_{λ}(|z|), and

1991 Mathematics Subject Classification. Primary 30C45, 30C50; Secondary 30C80.

Key words and phrases. pre-Schwarzian derivative, uniformly locally univalent, growth estimate, coefficient estimate.

|f(z)| ≤ F_{λ}(|z|)

in the unit disk. Furthermore, if f is univalent then

−F_{λ}(−|z|) ≤ |f(z)| ≤ F_{λ}(|z|).

If the equality occurs in any of the above inequalities at some point z_{0} ≠ 0, then f must be a rotation of F_{λ}, i.e., f(z) = μ̄F_{λ}(μz) for a unimodular constant μ.

Corollary 2. For λ > 1 any f ∈ B(λ) satisfies the growth condition

f(z) = O(1 − |z|)^{1−λ}

as |z| → 1. On the other hand, for λ < 1, a function f ∈ B(λ) is always bounded with a uniform bound F_{λ}(1). Furthermore, if f is univalent, then f(𝔻) contains the disk {|z| < −F_{λ}(−1)}. This constant −F_{λ}(−1) is best possible for 0 ≤ λ ≤ 1.

We note that for λ ≤ 1/2 the function f ∈ B(λ) must be univalent (cf. [2], [3]).

In this article, we will present several consequences from the above estimates and mention explicit norm estimates for various classes of univalent functions (for the proofs, see [4]). The following are sample theorems.

Theorem 3. Let 0 ≤ λ < 1. Then any function f ∈ B(λ) is Hölder continuous of exponent 1 − λ on the unit disk.

Theorem 4. Suppose f ∈ B(λ) is univalent.

If λ < 1 then f ∈ H^{∞}.

If λ > 1 then f ∈ H^{p} for any 0 < p < 1/(λ − 1).

If λ = 1 then f ∈ BMOA.

Note that H^{∞} ⊂ BMOA ⊂ ∩_{0<p<∞}H^{p}.

Let I_{p}(r, f) denote the integral mean of f with exponent p ∈ ℝ:

I_{p}(r, f) = \frac{1}{2π} ∫_{0}^{2π} |f(re^{iθ})|^{p}dθ.

And, for λ > 0, we set

α(λ) = \frac{\sqrt{1 + 4λ^{2}} − 1}{2}.

Note that

\frac{λ^{2}}{λ + 1} < α(λ) < min \{λ^{2}, \frac{2λ^{2}}{2λ + 1}\} ≤ min\{λ^{2}, λ\}.

Then we have the following

Theorem 5. Let f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ be in B(λ). Then, for any ε > 0 and a real number p, we have I_{p}(r, f′) = O(1 − r)^{−α(|p|λ)−ε}, in particular, a_{n} = O(n^{α(λ)−1+ε}).

Note that the extremal function F_{λ} has coefficients whose growth order is equivalent to n^{λ−2}.

The following is due to S. Yamashita. (The case of strongly starlike functions was first shown by [5].)

Theorem A (Yamashita [8]). Let 0 ≤ α < 1 and f ∈ S.

If f is starlike of order α, i.e., Re(zf′(z)/f(z)) > α, then ∥T_{f}∥ ≤ 6 − 4α.

If f is convex of order α, i.e., Re(1 + zf′′(z)/f′(z)) > α, then ∥T_{f}∥ ≤ 4(1 − α).

If f is strongly starlike of order α, i.e., arg(zf′(z)/f(z)) < πα/2, then ∥T_{f}∥ ≤ M(α) + 2α, where M(α) is a specified constant depending only on α satisfying 2α < M(α) < 2α(1 + α).

All of the bounds are sharp.

Finally we state general and useful principles for estimation of the norm of T_{f}. The following one always generates a sharp result for fixed g. The idea is due to Littlewood.

Theorem 6 (Subordination Principle I). Let g ∈ B be given. For f ∈ A, if f′ is subordinate to g′ then we have ∥T_{f}∥ ≤ ∥T_{g}∥. In particular, f is uniformly locally univalent on the unit disk.

We can also show the next result.

Theorem 7 (Subordination Principle II). Let g ∈ B be given. For f ∈ A, if zf′(z)/f(z) is subordinate to g′ then we have

∥T_{f}∥ ≤ sup_{z∈𝔻}(1 − |z|^{2}) (|\frac{g′(z) − 1}{z}| + |T_{g}(z)|)

≤ sup_{z∈𝔻}(1 − |z|^{2}) |\frac{g′(z) − 1}{z}| + ∥T_{g}∥.

References

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2. J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23–43.

3. J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math. 354 (1984), 74–94.

4. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, preprint, 1998.

5. T. Sugawa, On the norm of the pre-Schwarzian derivatives of strongly starlike functions, to appear in Ann. Univ. Mariae Curie-Skłodowska, Sectio A, 1998.

6. S. Yamashita, Banach spaces of locally schlicht functions with the Hornich operations, Manuscripta Math. 16 (1975), 261–275.

7. , Almost locally univalent functions, Monatsh. Math. 81 (1976), 235–240.

8. , Norm estimates for function starlike or convex of order alpha, to appear in Hokkaido Math. J., 1998.

9. I. V. Zhuravlev, Model of the universal Teichmüller space, Siberian Math. J. 27 (1986), 691–697.

Department of Mathematics, Yeungnam University, 214-1 Daedong, Gyongsan 712-749, Korea

  E-mail address: kimyc@ynucc.yeungnam.ac.kr

Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

  E-mail address: sugawa@kusm.kyoto-u.ac.jp