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

YONG CHAN KIM AND TOSHIYUKI SUGAWA

Abstract. In this note, we shall give a sharp distortion estimate for a uniformly locally univalent holomorphic function on the unit disk in terms of the norm of pre-Schwarzian derivative. As applications, we shall investigate the growth of coefficients and integral means of such a function and mention a connection with Hardy spaces. We also give norm estimates for typical classes of univalent functions.

1. Introduction

We will say that a holomorphic function f on the unit disk 𝔻 is 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 fixed positive number ρ. In particular, a holomorphic universal covering map of a plane domain D is uniformly locally univalent if and only if the boundary of D is uniformly perfect (see [17] or [23]). Also it is well-known (cf. [25]) 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 semi-norm of the function log f′. Remark that a holomorphic function f is locally univalent at the point z if and only if T_{f} = f′′/f′ is a well-defined holomorphic function near z. Roughly speaking, the quantity T_{f} measures the deviation of f from orientation-preserving similarities (non-constant linear functions). In the following, it is sometimes essential to consider the semi-norm

∥T_{f}∥_{0} = \overline{lim}_{|z|→1−0}(1 − |z|^{2})|T_{f}(z)| = 2 \overline{lim}_{|z|→1−0}(1 − |z|)|T_{f}(z)|

instead of ∥T_{f}∥. And, it is usually much easier to calculate ∥T_{f}∥_{0} than ∥T_{f}∥. We note that ∥T_{f}∥_{0} ≤ ∥T_{f}∥ always holds. A non-constant analytic function f on the unit disk is said to be almost uniformly locally univalent if and only if ∥T_{f}∥_{0} < ∞. For general properties of almost uniformly locally univalent functions, the reader may consult the lecture note [26] written by S. Yamashita.

Date: March 30, 1999.

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

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

The first named author was supported by the Korea Research Foundation(1998-015-D00019) and TGRC-KOSEF. The second named author was partially supported by the Ministry of Education, Grantin-Aid for Encouragement of Young Scientists, 9740056.

In this note, we will investigate the growth of various quantities for a uniformly locally univalent function in terms of the norm of pre-Schwarzian derivative. 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 ([24]).

For a non-negative real number λ we set

B(λ) = {f ∈ A; ∥T_{f}∥ ≤ 2λ},

here the factor 2 is due to only some technical reason. The functions in B(λ) can be characterized as the following.

Proposition 1.1. Let a non-negative constant λ be given. A locally univalent function f ∈ A belongs to B(λ) if and only if for any pair of points z_{1}, z_{2} in 𝔻 it holds that

|g(z_{1}) − g(z_{2})| ≤ 2λd_{𝔻}(z_{1}, z_{2}), (1.1)

where g(z) = log f′(z) and d_{𝔻}(z_{1}, z_{2}) = tanh^{−1} |\frac{z_{1}−z_{2}}{1−z̄_{1}z_{2}} | stands for the hyperbolic distance between z_{1} and z_{2} in the unit disk 𝔻.

Proof. First of all, note that we can take a holomorphic branch g of log f′ for a locally univalent holomorphic function f on the unit disk. The “only if ” part is shown by integrating the inequality |g′(z)| = |T_{f}(z)| ≤ 2λ/(1 − |z|^{2}) along the hyperbolic geodesic joining z_{1} and z_{2}. The “if ” part directly follows from the observation:

lim_{z′→z} \frac{|g(z′) − g(z)|}{d_{𝔻}(z′, z)} = (1 − |z|^{2})|g′(z)|.

The following theorem is significant in connection with univalent function theory.

Theorem A (Becker and Pommerenke [3], [4]). The set S of normalized univalent holomorphic functions on the unit disk is contained in B(3) and contains B(\frac{1}{2}). The result is sharp.

We note that the Schwarzian derivative S_{f} of f can be written as S_{f} = (T_{f})′ − (T_{f})^{2}/2. Thus the space B has a close connection with (the Bers embedding of) the universal Teichmüller space T , which is defined as the set of Schwarzian derivatives of those functions in S which can be quasiconformally extended to the Riemann sphere. Note that T is a contractible bounded domain in the complex Banach space B_{2} consisting of all holomorphic functions φ in the unit disk with finite norm ∥φ∥_{B_{2}} = sup_{z∈𝔻}(1 − |z|^{2})^{2}|φ(z)| and that {φ ∈ B_{2}; ∥φ∥_{B_{2}} < 2} ⊂ T ⊂ {φ ∈ B_{2}; ∥φ∥_{B_{2}} < 6}. Especially, it is expected to be useful when considering the Bers boundary of the Teichmüller spaces since the quantity T_{f} is much easier to treat than S_{f} in some cases. In fact, the space T_{1} := {T_{f}; f ∈ S has a quasiconformal extension to the Riemann sphere} can be regarded as a model of the universal Teichmüller space (cf. [1] and [29]). By the relation between S_{f} and T_{f}, we have the estimate ∥S_{f}∥_{B_{2}} ≤ C∥T_{f}∥ + ∥T_{f}∥^{2}/2, where C is an absolute constant. At least, we can take C = 4 (see [9]). On the other hand, as is stated in [7], the inequality ∥T_{f}∥ ≤ ∥S_{f}∥_{B_{2}} holds for a normalized function f in the Nehari class, i.e., for a function f ∈ A with f′′(0) = 0 such that ∥S_{f}∥_{B_{2}} ≤ 2 (see also [7]).

Here, as a result in this direction, we mention the following.

Corollary . For a constant k ∈ [0, 1), let S_{k} be the subset of S consisting of those functions which can be extended to k-quasiconformal self-mappings of the Riemann sphere ℂ̂. Then, we have

B(k/2) ⊂ S_{k}.

This implication is easily obtained by the λ-lemma (see, for example, [18, p.121]). This already appeared (implicitly) in the paper [3] of Becker.

Now we briefly explain the structure of this note. In Section 2, we state sharp growth and distortion theorems for the class B(λ). Those are simple analogues of the results of their paper [6], in which M. Chuaqui and B. Osgood obtained sharp growth, distortion and covering theorems and an estimate of Hölder continuity for normalized functions in the Nehari class in terms of the Nehari norm of Schwarzian derivatives.

As applications of those theorems, Section 3 discusses the Hölder continuity, growth of coefficients and integral means, and a connection with Hardy spaces for the class B(λ).

Section 4 is devoted to explicit estimates of the norm of pre-Schwarzian derivatives for typical classes of univalent functions. To this end, we will employ the subordination method.

Acknowledgements. A part of this work was carried out during authors’ visit to Fukuoka University, February 1998. They sincerely thank Fukuoka University, especially, Professor Megumi Saigo for inviting them.

2. Growth estimate for the class B(λ)

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 remark that F_{λ} ∈ A can be defined for any complex number λ and satisfies T_{F_{λ}} = 2λ(1−z^{2})^{−1}, thus ∥T_{F_{λ}}∥ = 2|λ|. F_{λ} may provide an example of a function with small pre-Schwarzian norm which does not belong to typical classes of univalent functions when λ is sufficiently small and λ ∉ ℝ.

In practice, it is important to know the univalence of F_{λ}.

Lemma 2.1. For a non-negative number λ, the function F_{λ} is univalent in the unit disk if and only if 0 ≤ λ ≤ 1.

Proof. First, we compute the Schwarzian derivative S_{F_{λ}} of F_{λ}. Then, we have

sup_{z∈𝔻}(1 − |z|^{2})^{2}|S_{F_{λ}}(z)| = sup_{z∈𝔻}(1 − |z|^{2})^{2} \frac{2λ|2z − λ|}{|1 − z^{2}|^{2}} = 2λ(λ + 2).

In particular, if 1 < λ, then 2λ(λ + 2) > 6, thus the Nehari-Kraus theorem implies that F_{λ} is not univalent.

On the other hand, if 0 ≤ λ ≤ 1, we have ReF′_{λ}(z) > 0 in the unit disk, hence the Noshiro-Warschawski theorem ensures the univalence of F_{λ} in this case.

The following result is elementary and may be known though we are unable to locate a reference. So, we shall include the proof because of its importance for our aim.

Theorem 2.2 (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 (2.1)

|f(z)| ≤ F_{λ}(|z|) (2.2)

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

−F_{λ}(−|z|) ≤ |f(z)| ≤ F_{λ}(|z|). (2.3)

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.

Proof. Applying Proposition 1.1 in the case of z_{1} = z and z_{2} = 0, we see

| log f′(z)| ≤ λ log \frac{1 + |z|}{1 − |z|}. (2.4)

Taking the real part of log f′, we obtain (2.1). And the integration of (2.1) yields (2.2). The inequality (2.3) can be shown by the same method as in the proof of the Koebe distortion theorem. The equality cases are obvious. (Note that the inequality (2.3) is sharp only for λ ≤ 1 by Lemma 2.1.)

Since ∫_{0}^{1}(\frac{1+t}{1−t})^{λ}dt < ∞ for λ < 1 and ∫_{0}^{r}(\frac{1+t}{1−t})^{λ}dt ≤ \frac{2^{λ}}{λ−1}(1 − r)^{1−λ} for λ > 1, we have the following

Corollary 2.3 (Growth and covering theorem). 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, any function f ∈ B(λ) is bounded with the uniform bound F_{λ}(1).

In both cases, 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.

By the same method, we have a similar conclusion as the first half in the above for a function f ∈ A with ∥T_{f}∥_{0} ≤ 2λ. In particular, if ∥T_{f}∥_{0} < 2, then f is bounded.

We note again that for λ ≤ 1/2 the function f ∈ B(λ) must be univalent. We also note that, for 0 ≤ λ ≤ 1, we have −F_{λ}(−1) ≥ −F_{1}(−1) = 2 log 2 − 1 = 0.38629 ⋯, therefore the result above is better than the Koebe one-quarter theorem.

Remark . By using the integral representation of the Gauss hypergeometric function (cf. Rainville [20] p.47, Theorem 16),

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

= ∑_{k=0}^{∞} \binom{λ}{k}z^{k} ∫_{0}^{1} t^{k}(1 − tz)^{−λ}dt

= ∑_{k=0}^{∞} \frac{Γ(λ + k)}{(k + 1)!Γ(λ)}z^{k}F (−λ, k + 1; k + 2; −z),

where F (a, b; c; z) denotes the Gauss hypergeometric function. Also, the values F_{λ}(1) and −F_{λ}(−1) can be expressed in terms of the Gauss hypergeometric function. For example, by [19] p.491,

−F_{λ}(−1) = ∫_{0}^{1} (\frac{1 − t}{1 + t})^{λ} dt = \frac{1}{λ + 1}F (1, λ; λ + 2; −1)

= \frac{1}{2^{λ}(λ + 1)}F (λ, λ + 1; λ + 2; 1/2)

= ∑_{k=0}^{∞} \frac{Γ(λ + k)}{k!(λ + k + 1)Γ(λ)2^{λ+k}} ,

which may also be rewritten in terms of the difference of two Digamma functions ([19], p.489, Eq.12) :

−F_{λ}(−1) = λ [ψ (\frac{λ + 1}{2}) − ψ (\frac{λ}{2})] − 1 (ψ(z) := \frac{Γ′(z)}{Γ(z)}) .

Similarly, we have F_{λ}(1) = λ[ψ(−λ/2) − ψ((1 + λ)/2)] − 1. It may be useful to note the following elementary estimate:

\frac{1}{(λ + 1)2^{λ}} < −F_{λ}(−1) < \frac{1}{λ + 1}.

In the above theorem, the case λ = 1 is critical. In this case, by Theorem 2.2, we can see that for f ∈ B(1)

|f(z)| ≤ F_{1}(|z|) = 2 log \frac{1}{1 − |z|} − |z|.

In particular, a function in B(1) need not be bounded (for instance, F_{1}). The next proposition gives a boundedness criterion for functions in B(1).

Proposition 2.4. If a holomorphic function f on the unit disk satisfies that

β(f) := \overline{lim}_{|z|→1−0} \{(1 − |z|^{2})|T_{f}(z)| − 2\} log \frac{1}{1 − |z|^{2}} < −2 (2.5)

then f is bounded. Here, the constant −2 in the right hand side is sharp.

Proof. By assumption, there exists a β < −2 such that the left-hand side in (2.5) is less than β. Thus, for some 0 < r_{0} < 1, (1 − |z|^{2})|T_{f}(z)| − 2 ≤ β/ log \frac{1}{1−|z|^{2}} , i.e.,

|T_{f}(z)| ≤ \frac{2}{1 − |z|^{2}} + \frac{β}{(1 − |z|^{2}) log \frac{1}{1−|z|^{2}}} (2.6)

for any z ∈ ℂ with r_{0} < |z| < 1. Here, we may choose r_{0} sufficiently close to 1 so that 1 − r_{0}^{2} < e^{−1}. Integrating the inequality (2.6), we see that, for |z| > r_{0},

| log f′(z)| ≤ log \frac{1 + |z|}{1 − |z|} + ∫_{r_{0}}^{|z|} \frac{βdt}{(1 − t^{2}) log \frac{1}{1−t^{2}}} + C_{1}

≤ log \frac{1 + |z|}{1 − |z|} + ∫_{r_{0}}^{|z|} \frac{βdt}{2(1 − t) log \frac{1}{2(1−t)}} + C_{1}

= log \frac{1 − |z|}{1 + |z|} + \frac{β}{2} log log \frac{1}{2(1 − |z|)} + C_{2},

where C_{1} and C_{2} are constants depending only on f and r_{0}. In particular, we have

|f′(z)| ≤ e^{C_{2}} \frac{1 + |z|}{1 − |z|} (log \frac{1}{2(1 − |z|)})^{β/2} .

Since β_{1}/2 < −1 the function \frac{1+t}{1−t}(log \frac{1}{2(1−t)})^{β/2} is integrable on the interval [r_{0}, 1). Thus f is bounded.

The sharpness follows from the example below.

Example 2.1. Let a constant β < 0 be given. Choose a constant c > 0 so that cβ+2 ≥ 0. Now we consider the function f ∈ A determined by

f′(z) = \frac{K}{1 − z} (1 + c log \frac{2}{1 − z} )^{β} ,

where K = (1 + c log 2)^{−β}. Then this function satisfies that ∥T_{f}∥ = 2. And moreover, f is bounded in the uint disk if and only if β < −1.

In fact, first observe that

T_{f}(z) = \frac{1}{1 − z} + \frac{cβ}{(1 − z)(1 + c log \frac{2}{1−z})} = \frac{1}{1 − z} [1 + \frac{β}{\frac{1}{c} + log \frac{2}{1−z}} ] .

By the fact that Re\frac{2}{1−z} > 1, one can conclude that Rew > \frac{1}{c} ≥ −β/2 and |Imw| < π/2, where w = \frac{1}{c} + log \frac{2}{1−z}. Noting that |1 + β/w|^{2} = 1 + β(2Rew + β)/|w|^{2} ≤ 1, one can see that |T_{f}(z)| ≤ \frac{1}{|1−z|} ≤ \frac{1}{1−|z|}. In particular, it holds that (1 − |z|^{2})|T_{f}(z)| ≤ 1 + |z| < 2. On the other hand, it is easy to see that lim_{x→1−0}(1 − x^{2})|T_{f}(x)| = 2, thus ∥T_{f}∥ = 2.

Next, we shall show that β(f) = 2β. Since |1 + β/w| = [1 + β(2Rew + β)/|w|^{2}]^{1/2} ∼ 1 + β(Rew + β/2)/|w|^{2} ∼ 1 + β/Rew ∼ 1 − β/ log |1 − z| as z → 1 and since the function t(1 + β/ log t) of t is monotonically increasing for sufficiently large t, we have

β(f) = \overline{lim}_{𝔻∋z→1}\{(1 − |z|^{2})|T_{f}(z)| − 2\} log \frac{1}{1 − |z|^{2}}

= \overline{lim}_{𝔻∋z→1}\{\frac{(1 − |z|^{2})}{|1 − z|} (1 + \frac{β}{log 1/|1 − z|}) − 2\} log \frac{1}{1 − |z|^{2}}

= \overline{lim}_{𝔻∋z→1}\{(1 + |z|) (1 + \frac{β}{log 1/(1 − |z|)}) − 2\} log \frac{1}{1 − |z|^{2}}

= \overline{lim}_{x→1−0} \{−(1 − x) log \frac{1}{1 − x^{2}} + (1 + x)β \begin{matrix}log & \frac{1}{1−x^{2}} \\ log & \frac{1}{1−x} \end{matrix} \} = 0 + 2β.

In particular, we can conclude that f is bounded if β < −1 by Proposition 2.4.

On the other hand, in the case that β ≥ −1, noting that ∫_{r_{0}}^{1} \frac{1}{1−x}(log \frac{1}{1−x})^{β} = ∞, we can directly see \overline{lim}_{x→1−0} f(x) = +∞, thus f is unbounded.

3. Applications

As applications of the results in the previous section, we will derive various properties of the functions in the class B(λ). We begin with the Hölder continuity of those functions. Recall the following fundamental fact due to Hardy-Littlewood.

Theorem B (cf. [8]). Let α be a constant such that 0 < α ≤ 1. A holomorphic function f on the unit disk is Hölder continuous of exponent α if and only if f′(z) = O(1 − |z|)^{α−1} as |z| → 1.

Combining this with Theorem 2.2, we have

Theorem 3.1. Let 0 ≤ λ < 1. Then any function f ∈ B(λ) is Hölder continuous of exponent 1 − λ on the unit disk.

Remarks . 1. We can directly see that |f(z_{1}) − f(z_{2})| ≤ \frac{C}{1−λ}|z_{1} − z_{2}|^{1−λ} for any pair of points z_{1}, z_{2} ∈ 𝔻, where C is an absolute constant, owing to the estimate ∫_{r}^{s}(\frac{1+t}{1−t})^{λ}dt ≤ \frac{2^{λ}}{1−λ}((1 − r)^{1−λ} − (1 − s)^{1−λ}) ≤ \frac{2^{λ}}{1−λ}(s − r)^{1−λ} for 0 < r < s < 1.

2. Chuaqui and Osgood proved in [6] that a normalized function f in the Nehari class is Hölder continuous with exponent \sqrt{1 − λ} where ∥S_{f}∥_{B_{2}} = 2λ. Their result is better than that obtained by combining the estimate ∥T_{f}∥ ≤ ∥S_{f}∥_{B_{2}} with the above theorem.

Second we consider coefficient estimates for the class B(λ). Let f(z) = z + a_{2}z^{2} + ⋯ ∈ B(λ). Then, by definition, |T_{f}(0)| ≤ 2λ, which implies |a_{2}| ≤ λ. Of course, this is sharp because the equality holds for the function F_{λ}. But, a function in B(λ) essentially different from F_{λ} may attain this maximum. For instance, consider the function f(z) = (e^{2λz} − 1)/2λ.

If the origin is a critical point of the function (1 − |z|^{2})|T_{f}(z)| then (T_{f})′(0) = 6a_{3} − (2a_{2})^{2} = 0 though this condition need not be sufficient for |a_{2}| = λ.

As for the growth of coefficients of a holomorphic function f(z) = a_{0} + a_{1}z + a_{2}z^{2} + ⋯ in the unit disk, it is convenient to consider the integral mean of exponent p ∈ ℝ:

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

In fact, we have the following elementary

Lemma 3.2. If I_{1}(r, f) = O(1 − r)^{−α} as r → 1 for a constant α ≥ 0, then we have a_{n} = O(n^{α}) as n → ∞.

Proof. Suppose that I_{1}(r, f) ≤ M(1−r)^{−α} for 0 ≤ r < 1. Then, for n > 1 and r = 1−1/n, it follows from Cauchy’s integral formula that

|a_{n}| = |\frac{1}{2π} ∫_{0}^{2π} f(re^{iθ})(re^{iθ})^{−n}dθ| ≤ r^{−n}I_{1}(r, f) ≤ Mr^{−n}(1 − r)^{−α}

= M (1 − \frac{1}{n})^{−n} n^{α} < \frac{eMn}{n − 1}n^{α}.

thus |a_{n}| < 2eMn^{α}.

In particular, for a function f(z) = z + a_{2}z^{2} + ⋯ in B(λ), by Theorem 2.2, we have I_{1}(r, f′) = O(1 − r)^{−λ}, thus |a_{n}| = O(n^{λ−1}) as n → ∞. Moreover if λ < 1 and if f is univalent, then f is bounded by Corollary 2.3, so

Area(f(𝔻)) = π (1 + ∑_{n=2}^{∞} n|a_{n}|^{2}) < ∞.

By this simple observation, we have a_{n} = o(n^{−1/2}) as n → ∞.

But we can improve the exponents in these trivial order estimates. We now explain this.

For λ > 0, we set

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

Noting α(λ) = 2λ^{2}/(\sqrt{1 + 4λ^{2}} + 1), then we have

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

We also note that

α(λ) = λ − \frac{1}{2} + \frac{1}{8λ} + O (\frac{1}{λ^{3}} ) (λ → ∞).

For this number, we have the next result.

Theorem 3.3. 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}).

This immediately follows from the next result.

Theorem C ([15, Lemma 5.3]). Let h be a holomorphic function in the unit disk such that

(1 − |z|) |\frac{h′(z)}{h(z)}| ≤ c (r_{0} ≤ |z| < 1)

for constants c > 0 and r_{0} < 1. Then, I_{p}(r, h) = O(1−r)^{−β}, where β = (\sqrt{1 + 4p^{2}c^{2}}−1)/2 and p ∈ ℝ.

We note that this is a consequence of the Fuchsian differential inequality:

I′′_{p}(r, h) ≤ \frac{p^{2}}{2π} ∫_{0}^{2π} |h(z)|^{p} |\frac{h′(z)}{h(z)}|^{2} dθ ≤ \frac{p^{2}c^{2}}{(1 − r)^{2}} I_{p}(r, h).

Moreover if f is univalent, we may have a better growth estimate for the coefficients. First we remind the reader of the following result due to Littlewood, Paley, Clunie, Pommerenke and Baernstein II (see [2], [18, Theorem 8.8] and [12, Theorem 3.7]).

Theorem D.Suppose that f(z) = z + a_{2}z^{2} + ⋯ ∈ S satisfies f(z) = O(1 − |z|)^{−α}. If 0.491 < α ≤ 2, then ∫_{0}^{2π} |f′(re^{iθ})|dθ = O(1 − r)^{−α} and a_{n} = O(n^{α−1}). If α = 0, in other words, if f is bounded, then ∫_{0}^{2π} |f′(re^{iθ})|dθ = O(1 − r)^{−0.491} and a_{n} = O(n^{0.491−1}).

In view of Corollary 2.3 we have the following result as a corollary.

Theorem 3.4. Let f(z) = z + a_{2}z^{2} + ⋯ ∈ S. If f ∈ B(λ) with 1.491 < λ ≤ 3, then it holds that a_{n} = O(n^{λ−2}) as n → ∞. This order estimate is best possible.

In order to see the sharpness, we may consider the function f(z) = (1 − z)^{1−λ} = 1 + a_{1}z + a_{2}z^{2} + ⋯ for 1 < λ. We note that f is univalent in the unit disk if 1 < λ ≤ 3. For this function, we can see that ∥T_{f}∥ = 2λ and a_{n} = Γ(λ + n − 1)/n!Γ(λ − 1) ∼ n^{λ−2} as n → ∞ by Stirling’s formula.

On the other hand, in the case that f is univalent with ∥T_{f}∥ < 3, the situation seems rather complicated. Given a holomorphic function f(z) = z + a_{2}z^{2} + ⋯ in the unit disk, let γ(f) denote the infimum of exponents γ such that a_{n} = O(n^{γ−1}) as n → ∞, i.e.,

γ(f) = \overline{lim}_{n→∞} \frac{log n|a_{n}|}{log n}.

And, for a subset X of A, we denote by γ(X) the supremum of {γ(f); f ∈ X}. As for γ(S_{b}), where S_{b} denotes the class of normalized bounded univalent functions in the unit disk, it has been shown ([5] and [14]) that 0.24 < γ(S_{b}) < 0.4886, and conjectured by Carleson and Jones that γ(S_{b}) = 0.25. We also remark that the growth of coefficients seems to involve an irregurality of the boundary of image under f when f is bounded and univalent (see [18, Chapter 10]) and, recently, Makarov and Pommerenke observed a remarkable phenomenon of phase transition of the functional γ(f) with respect to the Minkowski dimension of the boundary curve [14].

Now we turn to our case. Theorem 3.3 implies γ(B(λ)) ≤ α(λ). And the above example (1−z)^{1−λ} (or, − log(1−z) when λ = 1) shows λ−1 ≤ γ(B(λ)). By standard calculations, we can see that the extremal function F_{λ} also satisfies γ(F_{λ}) = λ − 1.

To construct an analytic function with curious boundary behaviour, the Hadamard gap series is often used (e.g., [18, §8.6]). Here, we present a simple example of such a kind to improve the above lower estimate of γ(B(λ)).

Example 3.1 (Gap series construction). Let q be a fixed integer greater than 1. We consider the function

g(z) = z + z^{q} + z^{q^{2}} + z^{q^{3}} + ⋯

in the unit disk, which can be characterized by the functional equation g(z) = z + g(z^{q}) with the initial condition g(0) = 0. We note that this is a Bloch function satisfying ∥g′∥ ≤ q/(q − 1) (cf. [18, §8.6]). Let t > 0 be a constant. Then the function h(z) = e^{tg(z)} = b_{0} + b_{1}z + b_{2}z^{2} + ⋯ obeys the functional equation h(z) = e^{tz}h(z^{q}). Thus the coefficients b_{n} are all positive and calculated by the relations

b_{kq+m} = ∑_{l=0}^{k} c_{lq+m}b_{k−l},

where c_{n} = t^{n}/n!. Letting m = 0, we have b_{kq} = c_{0}b_{k} + ⋯ + c_{kq}b_{0} > b_{k}. In particular, we know b_{q^{k}} > b_{q^{k−1}} > ⋯ > b_{1} = t. Therefore, we have \overline{lim} log b_{n}/ log n ≥ 0.

On the other hand, the function f ∈ A determined by f′ = h satisfies T_{f} = tg′, therefore ∥T_{f}∥ can be made arbitrarily small by letting t sufficiently small. This shows γ(B(λ)) ≥ 0 for any λ > 0.

Summarizing these observations, we have the next result.

Theorem 3.5. For any λ ∈ (0, ∞), we have

max\{0, λ − 1\} ≤ γ(B(λ)) ≤ α(λ) = \frac{\sqrt{1 + 4λ^{2}} − 1}{2}. (3.1)

In particular, γ(B(λ)) = O(λ^{2}) as λ → 0.

Remarks . 1. Recently, Chuaqui, Osgood and Pommerenke [7] proved that γ(B(λ)) ≥ cλ^{2} actually holds for some positive constant c when λ is sufficiently small. Their construction is rather technical and complicated, so our simple Example 3.1 seems still meaningful to be mentioned here.

2. More generally, by Theorem C, for any f ∈ A we have the estimate

γ(f) ≤ \frac{1}{2} (\sqrt{1 + ∥T_{f}∥_{0}^{2}} − 1) .

3. For 0 < λ ≤ 1/2, we note that α(λ) ≤ λ^{2} − 2λ^{4}/3 ≤ 5/24 = 0.2083 ⋯, because \sqrt{1 + x} < 1 + x/2 − x^{2}/(6 + 4\sqrt{2}) < 1 + x/2 − x^{2}/12 for 0 < x ≤ 1. Remark again that B(1/2) ⊂ S_{b}.

Next we consider the relationship between the class B(λ) and Hardy spaces. The following are fundamental results in the univalent function theory.

Theorem E (cf. [18]). Let β be a constant with 0 ≤ β ≤ 2. If a univalent function f ∈ S satisfies that f(z) = O(1 − |z|)^{−β} as |z| → 1, then the following holds.

For 0 < p < 1/β, we have f ∈ H^{p}. For 1/β < p, we have M_{p}(r, f) = O(1 − r)^{1/p−β} (r → 1).

Where M_{p}(r, f) denotes L^{p}-integral mean of f, i.e., M_{p}(r, f) = I_{p}(r, f)^{1/p}.

Theorem F (Pommerenke [16]). Let f be a univalent holomorphic function on the unit disk. Then, f ∈ BMOA if and only if f is Bloch, i.e., sup_{z∈𝔻}(1 − |z|^{2})|f′(z)| < ∞.

Combining these theorems with Theorem 2.2, we have the following results.

Theorem 3.6. Let f ∈ S and set ∥T_{f}∥ = 2λ.

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}.

Remark . Most of the above results can be extended to the case of p-valent, or more generally, mean p-valent functions with p < ∞ (see Hayman [12]).

We shall mention a connection with integral means for univalent functions. For a univalent function f ∈ S and a real number p, we set

β_{f}(p) = \overline{lim}_{r→1−0} \frac{log ∫_{0}^{2π} |f′(re^{iθ})|^{p}dθ}{log \frac{1}{1−r}} = \overline{lim}_{r→1−0} \frac{log I_{p}(r, f′)}{log \frac{1}{1−r}}.

The Brennan conjecture asserts that β_{f}(−2) ≤ 1 for every univalent holomorphic function f (cf. [18, Chapter 8]).

For f ∈ B(λ), as a corollary of Theorem 3.3, we have the next

Theorem 3.7. For f ∈ B(λ) amd p ∈ ℝ the inequality

β_{f}(p) ≤ α(|p|λ) = \frac{\sqrt{1 + 4p^{2}λ^{2}} − 1}{2}

holds. In particular, the Brennan conjecture is true for any univalent function f with ∥T_{f}∥ ≤ \sqrt{2}.

A similar statement can be found in [18, Exercise 8.3.4].

4. Norm estimates for various classes of univalent functions

In this section, we provide several norm estimates for well-known classes of univalent functions. These enable us to obtain growth and coefficient estimates for those classes, which agree with known results in many cases.

The following is due to S. Yamashita. (The case of strongly starlike functions was first shown by [22].)

Theorem G (Yamashita [28]). 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.

Remark . For the equality cases and more detailed and greatly general results, consult the paper [28] by S. Yamashita. For information about the constant M(α) see [22] or [28].

Now we state general and useful principles for estimation of the norm of T_{f}. A holomorphic function f on the unit disk is said to be weakly subordinate to another g if f can be written as f = g ∘ω, where ω is a holomorphic self-mapping of the unit disk. Furthermore if ω(0) = 0, the function f is said to be subordinate to g.

Remark that the Schwarz-Pick lemma implies that any holomorphic self-mapping ω of the unit disk satisfies

\frac{|ω′(z)|}{1 − |ω(z)|^{2}} ≤ \frac{1}{1 − |z|^{2}} (4.1)

for any point z ∈ 𝔻.

We also note that if g ∈ S, then f is weakly subordinate to g if and only if f(𝔻) ⊂ g(𝔻). The following always generates a sharp result for fixed g. The idea is due to Littlewood.

Theorem 4.1 (Subordination Principle I). Let g ∈ B be given. For a holomorphic function f in the unit disk, if f′ is weakly subordinate to g′ then we have ∥T_{f}∥ ≤ ∥T_{g}∥. In particular, f is uniformly locally univalent on the unit disk.

Proof. By assumption, there exists a holomorphic function ω : 𝔻 → 𝔻 such that f′ = g′ ∘ ω. Therefore, T_{f} = T_{g} ∘ ω · ω′. Thus (4.1) implies the following:

(1 − |z|^{2})|T_{f}(z)| = (1 − |z|^{2})|T_{g}(ω)||ω′| ≤ (1 − |ω|^{2})|T_{g}(ω)| ≤ ∥T_{g}∥,

which leads to the conclusion.

Remark . The analogous statement does not follow for the semi-norm ∥ · ∥_{0}. We also note that there exists an absolute constant c_{0} > 0 such that for any g ∈ B the inequality c_{0}∥T_{g}∥ ≤ sup_{f} ∥T_{f}∥_{0} ≤ ∥T_{g}∥ holds where the supremum is taken over all holomorphic functions f for which f′ is weakly subordinate to g′.

Actually, a single f is sufficient. In fact, take the holomorphic function f in the unit disk with f′ = g′ ∘ ω, where ω(z) = exp(−\frac{1+z}{1−z}) is a holomorphic universal covering map of the punctured disk 𝔻 \ {0}. The preimage of the circle |w| = e^{−a} under ω is a horocircle, say C_{a}, tangent to ∂𝔻 at 1. Since (1 − |z|^{2})|ω′(z)|/(1 − |ω(z)|^{2}) = a/ sinh a along that horocircle, we know

\overline{lim}_{C_{a}∋z→1}(1 − |z|^{2})|T_{f}(z)| = \frac{a}{sinh a} max_{|w|=e^{−a}}(1 − |w|^{2})|T_{g}(w)| = 2ae^{−a}M_{a},

where M_{a} = max_{|w|=e^{−a}} |T_{g}(w)|. In particular, we have 2ae^{−a}M_{a} ≤ \overline{lim}_{z→1}(1−|z|^{2})|T_{f}(z)| ≤ ∥T_{f}∥_{0}. When a ≥ 1 we have (1 − e^{−2a})M_{a} ≤ M_{a} ≤ M_{1} ≤ \frac{e}{2}∥T_{f}∥_{0}. When a < 1 we have (1 − e^{−2a})M_{a} ≤ \frac{sinh a}{a}∥T_{f}∥_{0} ≤ sinh 1∥T_{f}∥_{0} ≤ \frac{e}{2}∥T_{f}∥_{0}. Therefore we have ∥T_{g}∥ = sup_{a>0}(1 − e^{−2a})M_{a} ≤ \frac{e}{2}∥T_{f}∥_{0}.

As a typical application of the Subordination Principle, we exhibit the following.

Theorem 4.2. If f ∈ A satisfies that Ref′ > 0 on the unit disk, then ∥T_{f}∥ ≤ 2. The bound is sharp.

Remark . The Noshiro-Warschawski theorem says that such an f must be univalent.

Proof. The condition Ref′ > 0 is equivalent to the statement that f′ is subordinate to the function F′_{1}(z) = \frac{1+z}{1−z}. Thus we have ∥T_{f}∥ ≤ ∥T_{F_{1}}∥ = 2.

We note that f′ is a Gelfer function if Ref′ > 0, where a holomorphic function g on the unit disk with g(0) = 1 is called Gelfer when g(z) + g(w) ≠ 0 for all z, w ∈ 𝔻.

Therefore the next result can be viewed as a natural generalization of the above theorem.

Theorem 4.3. Suppose that f′ is a Gelfer function for an f ∈ A. Then we have ∥T_{f}∥ ≤ 2. This bound is sharp.

Proof. For a Gelfer function g(z) = f′(z) it is known to hold that

|\frac{g′(z)}{g(z)}| ≤ \frac{2}{1 − |z|^{2}}

(see [27]). Hence, the result immediately follows.

The next is a variant of the subordination principle.

Theorem 4.4 (Subordination Principle II). Let g ∈ B be given. For f ∈ A, if zf′(z)/f(z) is subordinate to g′ then we have

(4.2)  ∥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}∥. (4.3)

Proof. By assumption, there exists a holomorphic function ω : 𝔻 → 𝔻 with ω(0) = 0 such that zf′(z)/f(z) = g′(ω(z)). By taking logarithmic derivative, we have the following formula.

T_{f} = \frac{f′}{f} − \frac{1}{z} + \frac{g′′(ω)}{g′(ω)}ω′

= \frac{ω}{z} \frac{g′(ω) − 1}{ω} + T_{g}(ω)ω′.

From this, we can easily have the desired estimate.

The following is a simple application of this principle.

Theorem 4.5. If f ∈ A satisfies that |zf′(z)/f(z) − 1| < 1, then we have an estimate ∥T_{f}∥ ≤ 2.25. The equality holds if and only if f is a rotation of the function ze^{z}.

Remark . In this case, f satisfies Rezf′(z)/f(z) > 0 thus f is starlike, in particular, univalent in the unit disk.

Proof. We have only to apply the esitimate (4.2) with g(z) = z + z^{2}/2. Then, we have ∥T_{f}∥ ≤ sup(2+|z|−|z|^{2}) = 9/4, where the supremum is attained only by |z| = 1/2. Thus, if ∥T_{f}∥ = 9/4, then |ω| must be the constant 1, whence f is a rotation of ze^{z}. Conversely, it is clear that the function f(z) = ze^{μz} with |μ| = 1 satisfies ∥T_{f}∥ = 9/4.

Finally, we consider uniformly convex functions:

UCV = \{f ∈ S; Re (1 + (z − ζ)\frac{f′′(z)}{f′(z)}) ≥ 0, ∀z, ∀ζ ∈ 𝔻\} .

For the geometric meaning of this class, see [11]. Rønning gave a simple characterization for this class.

Theorem H (Rønning [21]). A function f ∈ A is uniformly convex if and only if zT_{f}(z) ∈ W for any z ∈ 𝔻, where W is the domain {w = u + iv; v^{2} < 2u + 1}.

We note that a conformal map g : 𝔻 → W with g(0) = 0 is given by

g(z) = \frac{2}{π^{2}} (log \frac{1 + \sqrt{z}}{1 − \sqrt{z}} )^{2} = \frac{8z}{π^{2}} (1 + \frac{z}{3} + \frac{z^{2}}{5} + \frac{z^{3}}{7} + ⋯)^{2} . (4.4)

Therefore, f ∈ A is uniformly convex if and only if zT_{f}(z) is subordinate to the function g, i.e., there exists a holomorphic function ω : 𝔻 → 𝔻 with ω(0) = 0 such that zT_{f}(z) = g(ω(z)). Since g has positive Taylor coefficients, we see that |zT_{f}(z)| ≤ g(|ω(z)|) ≤ g(|z|). Hence, we have

∥T_{f}(z)∥ ≤ sup_{0<x<1}(1 − x^{2})\frac{g(x)}{x} = sup_{0<t<∞} h(t),

where

h(t) = \frac{8t^{2}}{π^{2}} \frac{cosh t}{sinh^{2} t}

and \frac{1+\sqrt{x}}{1−\sqrt{x}} = e^{t}. By the logarithmic differentiation, we have

\frac{h′(t)}{h(t)} = \frac{2 sinh 2t − t(cosh 2t + 3)}{t sinh 2t} = \frac{N(t)}{t sinh 2t}.

Since N′′(t) = \frac{4(tanh 2t−t)}{cosh 2t} has the unique zero t_{0} in (0, ∞), the function N′(t) = 3(cosh 2t− 1) − 2t sinh 2t attains its maximum at t_{0}. Since N′(0) = 0 and N′(t) → −∞ as t → ∞, the function N′(t) has the unique zero t_{1} > t_{0} in (0, ∞). By exactly same reason, the function N(t) has the unique zero t_{2} > t_{1} in (0, ∞). Thus, h(t) assumes its maximum at the point t = t_{2}. By a numerical calculation, we have t_{2} = 1.6061152988 ⋯, and h(t_{2}) = 0.94774221287 · · · . Therefore, we summalize as follows.

Theorem 4.6. If f ∈ A is uniformly convex, then we have

∥T_{f}∥ ≤ h(t_{2}) = 0.94774 ⋯,

where the equality occurs only when f is a rotation of the function F ∈ A determined by T_{F} (z) = g(z)/z, where g is given by (4.4).

Remark . By the corollary of Theorem A, we see that a uniformly convex function can be extended to a h(t_{2})-quasiconformal self-homeomorphism of the Riemann sphere. As for quasiconformal extendability, we have a better estimate. In fact, from a recent result by Kanas and Rønning [13], for a uniformly convex function f the image of the function zf′(z)/f(z) lies in the domain

\{w; |w − 1| < Re w − \frac{1}{2}\} = \{w = u + iv; v^{2} < u − \frac{3}{4}\} ⊂ \{w; | arg w| < \frac{π}{6} \} .

Hence, we can conclude that every uniformly convex function is strongly starlike of order 1/3. By a theorem of Fait, Krzyż and Zygmunt [10], such a function can be extended to a \frac{1}{2}-quasiconformal automorphism of ℂ̂, since sin \frac{1}{3} \frac{π}{2} = \frac{1}{2}.

References

1. K. Astala and F. W. Gehring, Injectivity, the BMO norm and the universal Teichmüller space, J. Analyse Math. 46 (1986), 16–57.

2. A. Baernstein II, Coefficients of univalent functions with restricted maximum modulus, Complex Variables 5 (1986), 225–236.

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

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

5. L. Carleson and P. W. Jones, On coefficient problems for univalent functions and conformal dimension, Duke Math. J. 66 (1992), 169–206.

6. M. Chuaqui and B. Osgood, Sharp distortion theorems associated with the Schwarzian derivative, J. London Math. Soc. 48 (1993), 289–298.

7. M. Chuaqui, B. Osgood, and Ch. Pommerenke, On the coefficients of small univalent functions, Result. Math. 33 (1998), 79–86.

8. P. L. Duren, Theory of H^{p} Spaces, Academic Press, New York and London, 1970.

9. P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247–254.

10. M. Fait, J. G. Krzyż, and J. Zygmunt, Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv. 51 (1976), 279–285.

11. A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92.

12. W. K. Hayman, Multivalent Functions, Second edition, Cambridge University Press, 1994.

13. S. Kanas and F. Rønning, Uniformly starlike and convex functions and other related classes of univalent functions, Preprint, 1998.

14. N. G. Makarov and Ch. Pommerenke, On coefficients, boundary size and Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 22 (1997), 305–312.

15. Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

16. Ch. Pommerenke, Schlichite Funktionen und analytische Funktionen von beschränkten mittler Oszillation, Comment. Math. Helv. 52 (1977), 591–602.

17. , Uniformly perfect sets and the Poincaré metric, Ark. Math. 32 (1979), 192–199.

18. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, 1992.

19. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3, Gordon and Breach Science Publishers, 1990.

20. E. D. Rainville, Special Functions, Chelsea Publishing Company, New York, 1960.

21. F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196.

22. 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.

23. , Various domain constants related to uniform perfectness, Complex Variables (1998), 311–345.

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

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

26. , Lectures on Locally Schlicht Functions, Mathematical Seminar Notes, Tokyo Metropolitan Univ., 1977.

27. , Gelfer functions, integral means, bounded mean oscillation, and univalency, Trans. Amer. Math. Soc. 321 (1990), 245–259.

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

29. 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