The Norm Estimates of Pre-Schwarzian Derivatives of Spiral-like Functions

Yûsuke Okuyama^{∗}

Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Abstract For a constant β ∈ (−π/2, π/2), a normalized analytic function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ on the unit disk is said to be β-spiral-like if ℜ(e^{−iβ}zf′(z)/f(z)) > 0 for any point z in the unit disk. In this paper, for such a function f, we shall present the optimal estimate of the norm of f′′/f′.

1991 Mathematics Subject Classification. 30C45 Key words and phrases. spiral-like function, pre-Schwarzian derivative

1 Introduction

Let A denote the set of analytic functions f on the unit disk 𝔻 normalized so that f(0) = f′(0)−1 = 0. For a constant β ∈ (−π/2, π/2), a function f ∈ A is called β-spiral-like if f is univalent on 𝔻 and for any z ∈ 𝔻, the β-logarithmic spiral {f(z) exp(−e^{iβ}t); t ≥ 0} is contained in f(𝔻). It is equivalent to the condition that ℜ(e^{−iβ}zf′(z)/f(z)) > 0 in 𝔻. We denote by SP (β) the set of β-spiral-like functions. We call f_{β}(z) := z(1 − z)^{−2e^{iβ} cos β} ∈ SP (β) the β-spiral Koebe function. Note that SP (0) is the set of starlike functions and that f_{0}(z) = z(1 − z)^{−2} is the Koebe function. The β-spiral Koebe function conformally maps the unit disk onto the complement of the β-logarithmic spiral {f_{β}(−e^{−2iβ}) exp(−e^{iβ}t); t ≤ 0} in ℂ. For the known results about these classes of the functions, see, for example, [1].

For a locally univalent holomorphic function f, we define

T_{f} = \frac{f′′}{f′} and S_{f} = (T_{f})′ − \frac{1}{2}(T_{f})^{2},

∗E-mail; okuyama@kusm.kyoto-u.ac.jp

which are said to be the pre-Schwarzian derivative (or nonlinearity) and the Schwarzian derivative of f, respectively. For a locally univalent function f in 𝔻, we define the norms of T_{f} and S_{f} by

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

respectively.

As well as ∥S_{f}∥_{2}, the norm ∥T_{f}∥_{1} has a significant meaning in the theory of Teichmüller spaces. For example, see [9], [3] and [14]. On the other hand, there is a deep relation between the boundary of the universal Teichmüller space and such selfsimilar quasiarcs as logarithmic spirals ([2], [4], and [7]).

In the present paper, we shall give the best possible estimate of the norms of pre-Schwarzian derivatives for the class SP (β).

Main Theorem 1. For any f ∈ SP (β), where β ∈ (−π/2, π/2), we have the following.

I) In the case |β| ≤ π/3, we have

∥T_{f}∥_{1} ≤ ∥T_{f_{β}}∥_{1} = 2|2 + e^{2iβ}|.  (1)

II) In the case |β| > π/3, we have ∥T_{f}∥_{1} ≤ ∥T_{f_{β}}∥_{1}, where

∥T_{f_{β}}∥_{1} = max_{0≤m≤ \frac{4}{3} sin |β|} 2m cos β (1 + \sqrt{\frac{m^{2} + 4 − 4m sin |β|}{m^{2} + 1 − 2m sin |β|}} ) and (2)

2|2 + e^{2iβ}| < ∥T_{f_{β}}∥_{1} < 2 (1 + \frac{4}{3} sin 2|β|) .  (3)

In particular, ∥T_{f_{β}}∥_{1} → 2 as |β| → π/2.

In both cases, the equality ∥T_{f}∥_{1} = ∥T_{f_{β}}∥_{1} holds if and only if f is a rotation of the β-spiral Koebe function, i.e., f(z) = (1/ε)f_{β}(εz) for some |ε| = 1.

From the proof, if |β| ≤ π/3, the function (1−|z|^{2})|T_{f_{β}}(z)| does not attain its supremum in 𝔻. However if |β| > π/3, it does since

max_{∂𝔻∋z_{0}} lim sup_{𝔻∋z→z_{0}}(1 − |z|^{2})|T_{f_{β}}(z)| = 2|2 + e^{2iβ}| < ∥T_{f_{β}}∥_{1}.

This phenomenon of phase transition seems to be quite interesting. Remark. Clearly, the β-spiral Koebe function f_{β} converges to id_{𝔻} (which is bounded) locally uniformly on 𝔻 as |β| → π/2 but does not converge to it with respect to the norm ∥ · ∥_{1} since lim_{|β|→π/2} ∥T_{f_{β}}∥_{1} = 2. On the other hand, it is known that f ∈ A is bounded if ∥T_{f}∥_{1} < 2. Thus the value 2 is the least one of the norms of unbounded f ∈ A.

We would also like to mention the related works about norm estimates of pre-Schwarzian derivatives in other classes of A by Shinji Yamashita [12] and Toshiyuki Sugawa [10].

Theorem 1.1. Let 0 ≤ α < 1 and f ∈ A.

If f is starlike of order α, i.e., ℜ(zf′(z)/f(z)) > α, then ∥T_{f}∥_{1} ≤ 6 − 4α. If f is convex of order α, i.e., ℜ(1 + zf′′(z)/f′(z)) > α, then ∥T_{f}∥_{1} ≤

4(1 − α).

If f is strongly starlike of order α, i.e., arg(zf′(z)/f(z)) < πα/2, then ∥T_{f}∥_{1} ≤ 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.

On the other hand, we also obtain the estimate of the norms of Schwarzian derivatives of β-spiral-like functions.

Main Theorem 2. Assume |β| < π/2. For any f ∈ SP (β), ∥S_{f}∥_{2} ≤ ∥S_{f_{β}}∥_{2} = 6.

In Theorem 2, the extremality of f_{β} is trivial since the Kraus–Nehari theorem says that ∥S_{f}∥_{2} ≤ 6 for any univalent f ∈ A.

The proof of Theorems 1 and 2 will be given in Section 2 and 3. Knowing the norm ∥T_{f}∥_{1} of f ∈ A enables us to estimate the growth of coefficients of f. For example, the following holds.

Theorem 1.2 (cf. [8]). Let (3/2) < λ ≤ 3. For f(z) = z + a_{2}z^{2} + a_{3}z^{3} + ⋯ ∈ A such that ∥T_{f}∥_{1} ≤ 2λ, it holds that a_{n} = O(n^{λ−2}) as n → +∞. This order estimate is best possible.

We shall also remark on the sharp order estimate of coefficients of f ∈ SP (β) in Section 4.

2 Proof of Theorem 1

Let f ∈ A be a β-spiral-like function. We set p(z) = P_{f}(z) = zf′(z)/f(z). Then, by assumption, p is a holomorphic function on 𝔻 satisfying p(0) = 1 and p(𝔻) ⊂ \{w ∈ ℂ; −\frac{π}{2} + β < arg w < \frac{π}{2} + β\} =: ℍ_{β}. The univalent map q(z) = (1 + ze^{2iβ})/(1 − z) on 𝔻 satisfies q(0) = 1 and q(𝔻) = ℍ_{β}. Then p is subordinate to q, i.e., there exists a holomorphic function ω = ω_{f} : 𝔻 → 𝔻 with ω(0) = 0 such that

p = q ∘ ω = \frac{1 + ωe^{2iβ}}{1 − ω} .  (4)

We note that, for |ε| = 1, f(z) = (1/ε)f_{β}(εz) if and only if ω(z) = εz. By the logarithmic differentiation of (4), we have

T_{f}(z) =\frac{f′′(z)}{f′(z)} = c\frac{\frac{ω}{z}(1 + ωe^{2iβ}) + ω′}{(1 − ω)(1 + ωe^{2iβ})}, thus
(1 − |z|^{2})T_{f}(z) =c\frac{(1 − |z|^{2})\{\frac{ω}{z}(2 + ωe^{2iβ}) + (ω′ − \frac{ω}{z})\}}{(1 − ω)(1 + ωe^{2iβ})}.

Here we set c := e^{2iβ} + 1. Setting ω = id_{𝔻}, we also have

T_{f_{β}}(z) = c\frac{2 + ze^{2iβ}}{(1 − z)(1 + ze^{2iβ})} and  (5)
(1 − |z|^{2})T_{f_{β}}(z) = c\frac{(1 − |z|^{2})(2 + ze^{2iβ})}{(1 − z)(1 + ze^{2iβ})}.  (6)

We can easily see that max_{∂𝔻∋z_{0}} lim sup_{𝔻∋z→z_{0}}(1 − |z|^{2})|T_{f_{β}}(z)| = 2|2 + e^{2iβ}|.

By the Schwarz-Pick lemma for ω/z, we obtain (1 − |z|^{2})|zω′ − ω| ≤ |z|^{2} − |ω|^{2}. So we can estimate as

(1 − |z|^{2})|T_{f}(z)| ≤\frac{|ω|(1 − |z|^{2})}{|z|(1 − |ω|^{2})} · |c|\frac{(1 − |ω|^{2})|2 + ωe^{2iβ}|}{|1 − ω||1 + ωe^{2iβ}|}

+ \frac{|z|^{2} − |ω|^{2}}{|z|(1 − |ω|^{2})} · |c|\frac{1 − |ω|^{2}}{|1 − ω||1 + ωe^{2iβ}|}

=\frac{|2 + ωe^{2iβ}||ω|(1 − |z|^{2}) + (|z|^{2} − |ω|^{2})}{|2 + ωe^{2iβ}||z|(1 − |ω|^{2})}(1 − |ω|^{2})|T_{f_{β}}(ω)|.

To show ∥T_{f}∥_{1} < ∥T_{f_{β}}∥_{1} for SP (β) ∋ f with |ω′(0)| < 1, we show the following.

Lemma 2.1. Let ω : 𝔻 → 𝔻 be a holomorphic function with ω(0) = 0 and |ω′(0)| < 1. For any set U ⊂ 𝔻 with \overline{ω(U)} ∌ −e^{−2iβ}, there exists a positive constant C < 1 such that

D(z) := \frac{|2 + ωe^{2iβ}||ω|(1 − |z|^{2}) + (|z|^{2} − |ω|^{2})}{|2 + ωe^{2iβ}||z|(1 − |ω|^{2})} ≤ C (z ∈ U).

From the above lemma, we can conclude the following immediately.

Corollary 2.1. Let f ∈ SP (β) not be a rotation of f_{β}. For any set U ⊂ 𝔻 with \overline{ω_{f}(U)} ∌ −e^{−2iβ},

sup_{z∈U}(1 − |z|^{2})|T_{f}(z)| < ∥T_{f_{β}}∥_{1}

In particular, we can show the essentially unique extremality of f_{β} on some condition. Corollary 2.2. Let f ∈ SP (β) not be a rotation of f_{β}. If \overline{ω(𝔻)} ∌ −e^{−2iβ}, then ∥T_{f}∥_{1} < ∥T_{f_{β}}∥_{1}. Proof. We can take 𝔻 itself as such U in Corollary 2.1.

Proof of Lemma 2.1. We take such U as above.

Put

c_{1} := inf_{z∈U}(|2 + ω(z)e^{2iβ}| − 1) > 0.

For z ∈ U,

1 − D(z) = \frac{(|z| − |ω|)\{(|2 + ωe^{2iβ}| − 1)(1 + |z||ω|) + (1 − |z|)(1 − |ω|)\}}{|2 + ωe^{2iβ}||z|(1 − |ω|^{2})}

≥ \frac{(|z| − |ω|)\{c_{1}(1 + |z||ω|) + (1 − |z|)(1 − |ω|)\}}{6|z|(1 − |ω|)}

= \frac{1}{6}\{c_{1} \frac{1 + |z||ω|}{|z|}(1 − \frac{1 − |z|}{1 − |ω|}) + (1 − |\frac{ω}{z} |)(1 − |z|)\}.

In Yamashita [11] (p. 313, (6.8^{∗∗}a)), it is shown that for a holomorphic map ω : 𝔻 → 𝔻 with ω(0) = 0 which is not a rotation at the origin,

|ω(z)| ≤ |z|Q(|z|) < |z| (z ∈ 𝔻),  (7)

where

Q(x) = \frac{x^{2} + Bx + A}{Ax^{2} + Bx + 1} (0 ≤ x ≤ 1),

A = |ω′(0)| < 1 and

B = \frac{|ω′′(0)|}{2(1 − |ω′(0)|)} ≤ 1 + A < 2.

From this, it follows that

1 − D(z) ≥ \frac{1}{6}\{c_{1}(1 − \frac{1 − |z|}{1 − |z|Q(|z|)}) + (1 − Q(|z|))(1 − |z|)\}  (8)

on U. It is easy to see that Q(x) is strictly increasing and (1−x)/(1−xQ(x)) is strictly decreasing in [0, 1]. Thus for z ∈ U,

1 − D(z) ≥ \begin{cases}\frac{1}{6}(1 − Q(|z|))(1 − |z|) ≥ \frac{1}{12} (1 − Q(\frac{1}{2})) & if |z| < \frac{1}{2}, \\ \frac{1}{6}c_{1}(1 − \frac{\frac{1}{2}}{1− \frac{1}{2} Q( \frac{1}{2} )}) ≥ \frac{1}{12}c_{1} (1 − Q(\frac{1}{2})) & if \frac{1}{2} ≤ |z| < 1 \end{cases}

≥ \frac{1}{12} min(1, c_{1}) (1 − Q(\frac{1}{2})) > 0.

Now the proof is completed.

We consider the case \overline{ω(𝔻)} ∋ −e^{−2iβ}. Then inf_{z∈𝔻}(|2 + ω(z)e^{2iβ}| − 1) = 0, so the inequality similar to (8):

1 − D(z) ≥ \frac{1}{6}(1 − Q(|z|))(1 − |z|) > 0

holds for z ∈ 𝔻. We obtain the following.

Lemma 2.2. Let f ∈ SP (β) not be a rotation of f_{β}. For z ∈ 𝔻,

(1 − |z|^{2})|T_{f}(z)| < (1 − |ω|^{2})|T_{f_{β}}(ω)| ≤ ∥T_{f_{β}}∥_{1}.  (9)

In particular, ∥T_{f}∥_{1} ≤ ∥T_{f_{β}}∥_{1}. Moreover, if

max_{z_{0}∈∂𝔻} lim sup_{𝔻∋z→z_{0}}(1 − |z|^{2})|T_{f_{β}}(z)| = 2|2 + e^{2iβ}| < ∥T_{f_{β}}∥_{1},

this inequality is strict.

From now on, we turn our attention to the norm of T_{f_{β}}. If f ∈ SP (−β), then g(z) := \overline{f(\overline{z})} ∈ SP (β) and ∥T_{f}∥_{1} = ∥T_{g}∥_{1}, so we can assume β ≥ 0 without any loss of generality.

We consider the conformal automorphism z ↦ w = h(z) of 𝔻 with h(1) = 1, h(−e^{−2iβ}) = −1 and h(ie^{−iβ}) = i. This is given by the relation

\frac{w − 1}{w + 1} = e^{−iβ} \frac{z − 1}{z + e^{−2iβ}} .  (10)

By the Schwarz-Pick lemma, we have

1 − |z|^{2} = (1 − |w|^{2}) |\frac{dz}{dw}| .

Differentiating (10), we have

|\frac{dz}{dw}| = \frac{2|z + e^{−2iβ}|^{2}}{|c||w + 1|^{2}} .

From them, it follows that

1 − |z|^{2} = (1 − |w|^{2})\frac{2|z + e^{−2iβ}|^{2}}{|c||w + 1|^{2}} .

From (10), we also have

|1 − z| = |z + e^{−2iβ}| |\frac{w − 1}{w + 1}| .

Thus

(1 − |z|^{2})|T_{f_{β}}(z)| = |c|\frac{(1 − |z|^{2})|z + 2e^{−2iβ}|}{|1 − z||1 + ze^{2iβ}|} = 2\frac{1 − |w|^{2}}{|w^{2} − 1|} · |z + 2e^{−2iβ}|.

Since (1 − |w|^{2})/|w^{2} − 1| ≤ 1, we have (1 − |z|^{2})|T_{f_{β}}(z)| < 2|2 + e^{2iβ}| on {z ∈ 𝔻; |z + 2e^{−2iβ}| < |1 + 2e^{−2iβ}|}. In the case β = 0, it coincides the whole 𝔻. Therefore it is sufficient to consider the only case β > 0. For the estimate of (1 − |z|^{2})|T_{f_{β}}(z)| on {z ∈ 𝔻; |z + 2e^{−2iβ}| ≥ |1 + 2e^{−2iβ}|}, we use some geometric argument.

Noting that |w^{2} − 1|^{2} = (1 − |w|^{2})^{2} + 4(ℑw)^{2}, we can see that the circular arc C_{1} passing through the three points ±1 and ki (|k| < 1) in the w-plane is the following:

\frac{1 − |w|^{2}}{|w^{2} − 1|} = \frac{1 − k^{2}}{1 + k^{2}} .

So C_{1} is the level curve of (1 − |w|^{2})/|w^{2} − 1|. Put C_{2} = h^{−1}(C_{1}). Since C_{2}

1 ie^{−iβ} −e^{−2iβ} −2e^{−2iβ} 1 −1  i−i  ki  0 s 0  h w-plane  z-plane C_{1}  C_{2}  z(m) h^{−1}(ki)

Figure 1: the level curve C_{1} and C_{2}

is the circle passing through the three points

1, − e^{−2iβ} and h^{−1}(ki) = ie^{−iβ} · \frac{(1 − ki)e^{−iβ} − (1 + ki)}{(k − i)e^{−iβ} − (k + i)},

we can calculate the center s and the radius r of C_{2} :

s =\frac{ie^{−iβ}(k^{2} − 1)}{2k cos β + (k^{2} − 1) sin β} and  (11)
r =\frac{(k^{2} + 1) cos β}{|2k cos β + (k^{2} − 1) sin β|}.  (12)

Putting m := \frac{k^{2}−1}{2k cos β+(k^{2}−1) sin β}, we have s = m · ie^{−iβ}. On the level curve C_{2}, |z + 2e^{−2iβ}| takes the maximum at the point z(m) in Figure 1, which is the intersection of the circular arc C_{2} and the straight line passing through −2e^{−2iβ} and s = m · ie^{−iβ}. Therefore

(1 − |z|^{2})|T_{f_{β}}(z)| ≤ 2\frac{1 − k^{2}}{1 + k^{2}} (|s + 2e^{−2iβ}| + r)

on C_{2}.

ie^{−iβ} −e^{−2iβ} −2e^{−2iβ}  0 s z-plane C_{2}  z(m_{0}) = 1

Figure 2: the level curve C_{2} in the case m = m_{0}.

Since we are considering the case |z+2e^{−2iβ}| ≥ |1+2e^{−2iβ}|, we can assume 0 ≤ m ≤ m_{0} := \frac{4}{3} sin β. We note that if m = m_{0}, then C_{2} is tangential to the circular arc {z ∈ 𝔻; |z + 2e^{−2iβ}| = |1 + 2e^{−2iβ}|} and the tangent point between them is z(m_{0}) = 1, and that if m moves from 0 to m_{0}, the level curve C_{2} sweeps out {z ∈ 𝔻; |z + 2e^{−2iβ}| ≥ |1 + 2e^{−2iβ}|} (see Figure 2).

Noting that 2k cos β + (k^{2} − 1) sin β < 0 since m ≥ 0 and k^{2} ≤ 1, we also have r = \frac{1+k^{2}}{1−k^{2}} m cos β. It follows that

|s + 2e^{−2iβ}| = \sqrt{m^{2} + 4 − 4m sin β},
\frac{1 − k^{2}}{1 + k^{2}} = \frac{m cos β}{\sqrt{m^{2} + 1 − 2m sin β}} and for z ∈ C_{2},

(1 − |z|^{2})|T_{f_{β}}(z)| ≤ E(m) := 2m cos β (1 + \sqrt{\frac{m^{2} + 4 − 4m sin β}{m^{2} + 1 − 2m sin β}} ) .

We also note that z(m_{0}) = 1 ∈ ∂𝔻 and that z(m) ∈ {z ∈ 𝔻; |z + 2e^{−2iβ}| ≥ |1 + 2e^{−2iβ}|} for 0 ≤ m < m_{0}.

Here we consider the case 0 < β ≤ π/3. Noting that 2|2 + e^{2iβ}| = 2\sqrt{5 + 4 cos 2β} and E(m_{0}) = 4 sin 2β, we can see that E(m_{0}) ≤ 2|2 + e^{2iβ}| and the equality holds if and only if β = π/3.

The following holds.

Lemma 2.3. If 0 < β ≤ π/3, then E(m) ≤ E(m_{0}) for any 0 ≤ m ≤ m_{0}, and the equality holds if and only if m = m_{0}.

Proof. It is easy to see that E(0) < E(m_{0}) and that for 0 < m ≤ m_{0}, E(m) ≤ E(m_{0}) if and only if

(m − m_{0}){2m^{2} sin β + (1 − 8 sin^{2} β)m + 4 sin β} ≤ 0.

Putting

g(m) := 2m^{2} sin β + (1 − 8 sin^{2} β)m + 4 sin β,  (13)

we calculate as

g(m) = 2 sin β (m + \frac{1 − 8 sin^{2} β}{4 sin β} )^{2} − \frac{64 sin^{4} β − 48 sin^{2} β + 1}{8 sin β} and

m_{0} − (−\frac{1 − 8 sin^{2} β}{4 sin β} ) = \frac{3 − 8 sin^{2} β}{12 sin β} .

The following holds:

(i) If 0 < sin^{2} β ≤ \frac{1}{8}, then g(m) > 0 for any 0 < m ≤ m_{0} from (13).

(ii) If \frac{1}{8} < sin^{2} β ≤ \frac{3}{8}, the same thing as the above holds since −\frac{64 sin^{4} β−48 sin^{2} β+1}{8 sin β} > 0.

(iii) If \frac{3}{8} < sin^{2} β ≤ \frac{3}{4}, then g(m) ≥ 0 for any 0 < m ≤ m_{0} since g(m) is decreasing in (0, m_{0}] and

g(m_{0}) = g(\frac{4}{3} sin β) = −\frac{64}{9} sin β(sin^{2} β − \frac{3}{4}) ≥ 0.

Thus E(m) ≤ E(m_{0}) for any 0 ≤ m ≤ m_{0} and the equality holds if and only if m = m_{0}.

Consequently it follows that for |β| ≤ π/3, (1 − |z|^{2})|T_{f_{β}}(z)| < 2|2 + e^{2iβ}| on 𝔻. Noting that (1 − |z|^{2})|T_{f_{β}}(z)| tends to 2|2 + e^{2iβ}| as z tends to 1 − 0 along the real axis, we can conclude that ∥T_{f_{β}}∥_{1} = 2|2 + e^{2iβ}| and that the function (1 − |z|^{2})|T_{f_{β}}(z)| does not attain its supremum in 𝔻.

Next we consider the case β > π/3. In this case, we can see ∥T_{f_{β}}∥_{1} is strictly larger than 2|2 + e^{2iβ}|. In fact, we have 0 ≤ 1/ sin β < m_{0} and E(1/ sin β) > 2|2 + e^{2iβ}|. Therefore from Lemma 2.2, we can also conclude that a rotation of f_{β} is a unique extremal function.

Moreover, for 0 ≤ m ≤ \frac{4}{3} sin β, we have a uniform estimate:

E(m) = 2m cos β (1 + |1 − \frac{ie^{−iβ}}{m − ie^{−iβ}} |)

< 2m cos β (2 + \frac{1}{|m − ie^{−iβ}|})

≤ \frac{8}{3} sin 2β + 2 cos β \frac{m}{|m − ie^{−iβ}|}

≤ 2 (1 + \frac{4}{3} sin 2β) .

Thus ∥T_{f_{β}}∥_{1} → 2 as β → π/2.

Finally, we will show that for |β| ≤ π/3, f_{β} is also the essentially unique extremal function in SP (β).

Let f ∈ SP (β) not be a rotation of f_{β}. Noting Corollary 2.2, we consider the only case that \overline{ω(𝔻)} ∋ −e^{−2iβ}. Put ε := |2 + e^{2iβ}| − 1 > 0. Noting that

lim sup_{𝔻∋z→−e^{−2iβ}}(1 − |z|^{2})|T_{f_{β}}(z)| = 2,

we can take the constant r > 0 such that (1 − |z|^{2})|T_{f_{β}}(z)| < 2 + ε on N := {z ∈ 𝔻; |z + e^{−2iβ}| < r}. We note that 2 + ε < 2|2 + e^{2iβ}| = ∥T_{f_{β}}∥_{1}. Next put M := ω^{−1}(N). From (9) in Lemma 2.2, we obtain the following:

sup_{z∈M}(1 − |z|^{2})|T_{f}(z)| ≤ sup_{z∈N}(1 − |z|^{2})|T_{f_{β}}(z)| ≤ 2 + ε < ∥T_{f_{β}}∥_{1}.

On the other hand, since \overline{ω(𝔻 \ M)} ∌ −e^{−2iβ}, we have

sup_{z∈𝔻\M}(1 − |z|^{2})|T_{f}(z)| < ∥T_{f_{β}}∥_{1}.

Combining both estimates, we can conclude ∥T_{f}∥_{1} < ∥T_{f_{β}}∥_{1}.

Now the proof of Theorem 1 is completed.

3 Proof of Theorem 2

From (5), it follows that

S_{f_{β}} =(T_{f_{β}})′ − \frac{1}{2}(T_{f_{β}})^{2}
= − c\frac{e^{2iβ}\{e^{2iβ}(e^{2iβ} − 1)z^{2} + 4(e^{2iβ} − 1)z + 6\}}{2(1 − z)^{2}(1 + ze^{2iβ})^{2}}.

So we also have

(1 − |z|^{2})^{2}|S_{f_{β}}(z)| = |c|\frac{(1 − |z|^{2})^{2}|e^{2iβ}(e^{2iβ} − 1)z^{2} + 4(e^{2iβ} − 1)z + 6|}{2|1 − z|^{2}|1 + ze^{2iβ}|^{2}}.

It follows easily that (1 − |z|^{2})^{2}|S_{f_{β}}(z)| → 6 as z → −e^{−2iβ} radially. By the Kraus-Nehari theorem, sup_{z∈𝔻} |S_{f_{β}}(z)|(1 − |z|^{2})^{2} ≤ 6. Therefore we obtain ∥S_{f_{β}}∥_{2} = 6 for any |β| < π/2.

4 Order estimate of the coefficients

Knowing the norm ∥T_{f}∥_{1} enables us to estimate the growth of coefficients of f (cf. [8]). However the sharp estimate of coefficients of f ∈ SP (β) has been already obtained by Zamorski [13] in 1960. We would like to remark that we can derive the sharp growth estimate of coefficients of f ∈ SP (β) from this.

Theorem 4.1 (Zamorski). If f(z) = z +a_{2}z^{2} +a_{3}z^{3} +⋯ is in SP (β) and |β| < π/2, then

|a_{n}| ≤ ∏_{k=1}^{n−1} |1 + \frac{e^{2iβ}}{k} |  (14)

for any n ≥ 2. The equality in (14) holds for some n ≥ 2 if and only if f is a rotation of the β-spiral Koebe function f_{β}.

Remark. This is also shown in terms of generalized spiral-like functions by C. Burniak, J. Stankiewicz and Z. Stankiewicz [6](1980).

Corollary 4.1. Let |β| < π/2 and f(z) = z +a_{2}z^{2} +a_{3}z^{3} +⋯ be a β-spiral-like function. Then it holds that

a_{n} = O(n^{cos 2β}) (n → +∞).  (15)

This order estimate is sharp.

Proof. From the inequality (14), we have that for |β| < π/2,

log |a_{n}| ≤\frac{1}{2} ∑_{k=1}^{n−1} log (1 + \frac{2 cos 2β}{k} + \frac{1}{k^{2}} )

=\frac{1}{2} ∑_{k=1}^{n−1} (\frac{2 cos 2β}{k} ) + O(1)
= cos 2β log n + O(1)

as n → +∞. Therefore we obtain the estimate (15).

Remark. In the case |β| < π/4, this is shown by Basgöze and Keogh in [5](1970).

Acknowledgements

The author would like to express his gratitude to Prof. Masahiko Taniguchi and Prof. Toshiyuki Sugawa for many valuable discussions and advices. This research was partially supported by JSPS Research Fellowships for Young Scientists.

References

[1] Ahuja, O. P. and Silverman, H. A survey on spiral-like and related function classes, Math. Chronicle, 20 (1991), 39–66.

[2] Astala, K. Selfsimilar zippers, Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 11 (1988), 61–73.

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

[4] Astala, K. and Gehring, F. W. Crickets, zippers and the Bers universal Teichmüller space, Proc. Amer. Math. Soc., 110 (1990), 675–687.

[5] Basgöze, T. and Keogh, F. B. The Hardy class of a spiral-like function and its derivative, Proc. Amer. Math. Soc, 26 (1970), 266–269.

[6] Burniak, C., Stankiewicz, J. and Stankiewicz, Z. The estimations of coefficients for some subclasses of spirallike functions, Demonstratio Math., 15 (1982), 427–441.

[7] Gehring, F. W. Spirals and the universal Teichmüller space, Acta Math, 141 (1978), 99–113.

[8] Kim, Y. C. and Sugawa, T. Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Sūrikaisekikenkyūsho Kōkyūroku, 1062 (1998), 34–46.

[9] Lehto, O. Univalent Functions and Teichmüller Spaces, Springer-Verlag (1987).

[10] Sugawa, T. On the norm of the pre-Schwarzian derivatives of strongly starlike functions, Ann. Univ. Mariae Curie-Skłodowska, Sectio A, 52 (1998), 149–157.

[11] Yamashita, S. The Pick version of the Schwarz lemma and comparison of the Poincaré densities, Ann. Acad. Sci. Fenn., 19 (1994), 291–322.

[12] Yamashita, S. Norm estimates for function starlike or convex of order alpha, Hokkaido Math, 28 (1999), 217–230.

[13] Zamorski, J. About the extremal spiral schlicht functions, Ann. Polon. Math, 9 (1961), 265–273.

[14] Zhuravlev, I. V. A model of the universal Teichmüller space, Siberian Math. J., 27 (1986), 691–697.