A REMARK ON LÖWNER’S THEOREM

TOSHIYUKI SUGAWA

Abstract. We give a refinement of Löwner’s inequality with argument of the equality case. To this end, we establish a complete univalence criterion for meromorphic functions of special type. We also give applications of the refinement.

1. Introduction

Let Σ be the set of univalent meromorphic functions F on Δ = {ζ ∈ ℂ : |ζ| > 1}∪{∞} of the form

F (ζ) = ζ + ∑_{n=0}^{∞} b_{n}ζ^{−n}, |ζ| > 1.

Gronwall’s area theorem

(1)  ∑_{n=1}^{∞} n|b_{n}|^{2} ≤ 1

for F ∈ Σ leads to many valuable results in the theory of univalent functions (see [2] for instance). We remark that equality holds in (1) precisely when the image F (Δ) is of full measure, namely, ℂ \ F (Δ) is of area zero.

A simple consequence of the area theorem is K. Löwner’s theorem [5, Satz V].

Theorem A (Löwner). Let F ∈ Σ. Then

|F′(ζ)| ≤ \frac{|ζ|^{2}}{|ζ|^{2} − 1}, |ζ| > 1,

where equality holds at a finite point ζ = ζ_{0} in Δ if and only if

F (ζ) = ζ + b_{0} − \frac{ζ_{0} − ζ̄_{0}^{−1}}{ζ̄_{0}ζ − 1} , |ζ| > 1

for a constant b_{0}.

We slightly improve this theorem as in the following.

Theorem 1. For F ∈ Σ, the inequality

(2)  |F′(ζ) − 1| ≤ \frac{1}{|ζ|^{2} − 1}

Date: 9 October, 2006, File: loewner.tex.

Key words and phrases. univalent function, pre-Schwarzian derivative.

The author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039.

holds for each ζ ∈ ℂ with |ζ| > 1. Moreover, equality holds at ζ = ζ_{0} if and only if

F (ζ) = ζ + b_{0} − \frac{ζ_{0} − ζ̄_{0}^{−1}}{ζ̄_{0}ζ − 1} , |ζ| > 1,

for a constant b_{0}.

Since |F′(ζ)| − 1 ≤ |F′(ζ) − 1|, Löwner’s theorem immediately follows from Theorem 1. Indeed, as will be seen later, Löwner’s estimation in [5] works for (2) as well. However, discussion for the equality case becomes more subtle in the proof of our result.

We have to mention a theorem of Goluzin (cf. [3, (12) in p. 133]), which is a special case of well-known Goluzin’s inequality.

Theorem B.

Re \{e^{2iα} log F′(ζ)\} ≤ |log F′(ζ)| ≤ log \frac{|ζ|^{2}}{|ζ|^{2} − 1}, |ζ| > 1,

for F ∈ Σ, −π/2 ≤ α ≤ π/2. Here, for each ζ = ζ_{0} and for each α, equality holds at the same time in the above inequalities when F maps Δ onto the extended plane slit along an arc of the curve {w : Re (e^{iα} log(w − F (ζ_{0}))) = C} for a constant C.

Indeed, this inequality implies (2) (see Remark in Section 3). Since the description of the equality case in Goluzin’s theorem is somewhat implicit, our theorem seems to have its own merit.

The present paper is organized as follows. In the second section, we give two elementary lemmas, which may be of independent interest. The proof of Theorem 1 is provided in Section 3. The last section is devoted to a few applications of Theorem 1.

2. Some lemmas

Denote by 𝔻 the unit disk {z ∈ ℂ : |z| < 1}. We now show the following elementary lemma.

Lemma 2. Let L(z) = (az + b)/(cz + d) be a Möbius transformation, where a, b, c, d ∈ ℂ with ad − bc ≠ 0. Then L maps the unit disk 𝔻 into itself if and only if

|ac̄ − bd̄| + |ad − bc| ≤ |d|^{2} − |c|^{2}.

Proof. First we assume that L(𝔻) ⊂ 𝔻. Then L^{−1}(∞) = −d/c ∈ Δ, and thus, |c| < |d|. Let α = c̄/d̄∈ 𝔻. We can then express L by

L(z) = A\frac{z + α}{1 + ᾱz} + B,

where

A = \frac{ad − bc}{d^{2}(1 − |c/d|^{2})}, B = \frac{bd̄− ac̄}{|d|^{2} − |c|^{2}} .

Since the Möbius transformation (z + α)/(1 + ᾱz) maps 𝔻 onto itself, the image L(𝔻) is the disk centered at B with radius |A|. Therefore, L(𝔻) ⊂ 𝔻 implies |A| + |B| ≤ 1, which is equivalent to the inequality in the assertion.

By tracing back the above argument, we prove the other way round, as well. □

Secondly, we give a characterization of univalence for meromorphic functions of a specific form.

Lemma 3. Let b_{0}, A and B be complex numbers with AB ≠ 0. Then the meromorphic function

F (ζ) = ζ + b_{0} + \frac{A}{Bζ − 1}

is univalent in Δ if and only if

(3)  |A + B̄ − B^{−1}| + |AB| ≤ |B|^{2} − 1.

Proof. Suppose that F (ζ) = F (ω) for some ζ, ω ∈ ℂ with ζ ≠ ω. A straightforward computation yields the relation (Bζ−1)(Bω−1) = AB, which is equivalent to ω = L(1/ζ), where

L(z) = \frac{(A − 1/B)z + 1}{−z + B}.

If L(𝔻) ⊂ 𝔻, then L(1/ζ) ∈ 𝔻 for each ζ ∈ Δ. Therefore, according to the above computaion, there is no other point ω ∈ Δ than ζ such that F (ζ) = F (ω). Conversely, if F is univalent in Δ, the point ω = L(1/ζ) may not belong to Δ \ {ζ} for each ζ ∈ Δ, in other words, |L(z)| ≤ 1 for each z ∈ 𝔻 except for the fixed points of the Möbius transformation 1/L(z). Since L(𝔻) is an open disk or an open half-plane in general, this implies L(𝔻) ⊂ 𝔻. In this way, we have shown that F is univalent in Δ if and only if L(𝔻) ⊂ 𝔻. We now apply Lemma 2 to deduce the assertion. □

3. Proof of Theorem 1

We follow the argument in [5] for the first part. Let F (ζ) = ζ + b_{0} + b_{1}/ζ + b_{2}/ζ^{2} + … be in Σ. Then

F′(ζ) = 1 − ∑_{n=1}^{∞} nb_{n}ζ^{−n−1}.

By the Cauchy-Schwarz inequality, we have

|F′(ζ) − 1| ≤ ∑_{n=1}^{∞} n|b_{n}||ζ|^{−n−1}

≤ \sqrt{∑_{n=1}^{∞} n|b_{n}|^{2} · ∑_{n=1}^{∞} n|ζ|^{−2n−2}}

= \sqrt{∑_{n=1}^{∞} n|b_{n}|^{2} · \frac{1}{(|ζ|^{2} − 1)^{2}}} .

We now use (1) to obtain (2).

Next we consider the equality case. We suppose now that equality holds in (2) for ζ = ζ_{0}. Then the sequences {b_{n}} and {ζ̄_{0}^{−n−1}} must be proportional, in other words, there exists a complex number a such that b_{n} = aζ̄_{0}^{−n} for n ≥ 1. Also, since equality must hold in (1), we have |a| = R − R^{−1}, where R = |ζ_{0}| > 1.

Then

F (ζ) = ζ + b_{0} + a ∑_{n=1}^{∞} ζ̄_{0}^{−n}ζ^{−n} = ζ + b_{0} + \frac{a}{ζ̄_{0}ζ − 1}.

For this function, clearly equality holds in (2) at ζ = ζ_{0}. It remains to see when this function F is univalent in Δ. We now apply Lemma 3 to see that F is univalent precisely when

|a + ζ_{0} − ζ̄_{0}^{−1}| + |a|R = |a + ζ_{0} − ζ̄_{0}^{−1}| + R^{2} − 1 ≤ R^{2} − 1,

which is equivalent to |a + ζ_{0} − ζ̄_{0}^{−1}| = 0. Therefore, F is univalent precisely if a = −(ζ_{0} − ζ̄_{0}^{−1}), which proves the equality part.

Remark. Inequality (2) also follows from Goluzin’s inequality in Thereom B. Indeed, for a fixed ζ, we set w = log F′(ζ). Then

|F′(ζ) − 1| = |e^{w} − 1| = |∫_{0}^{w} e^{z}dz|

≤ ∫_{0}^{|w|} e^{x}dx = e^{|w|} − 1.

Since Goluzin’s inequality yields e^{|w|} ≤ |ζ|^{2}/(|ζ|^{2} − 1), we obtain (2). Equality holds precisely when w = log(|ζ|^{2}/(|ζ|^{2} − 1)).

4. Remarks and applications

First we remark on the image of Δ under the extremal map

F_{ζ_{0}}(ζ) = ζ − \frac{ζ_{0} − ζ̄_{0}^{−1}}{ζ̄_{0}ζ − 1}

for ζ_{0} = Re^{iα}, R > 1. As is stated in [5], the image F_{ζ_{0}}(Δ) is the slit domain along a proper subarc of a circle of radius R. More precisely, we have the following.

Lemma 4. Let ζ_{0} = Re^{iα} ∈ Δ. Then F_{ζ_{0}}(Δ) = ℂ̂ \ C_{ζ_{0}}, where C_{ζ_{0}} is the closed circular arc which is contained in the circle {ζ : |ζ − (ζ_{0} − ζ̄_{0}^{−1})| = R}, has the endpoints at (1 ± i\sqrt{R^{2} − 1})/ζ̄_{0}, and contains the point −ζ̄_{0}^{−1}.

Proof. First observe that the image of the unit circle is cotained in the circle in the assertion. Indeed, since

F_{ζ_{0}}(ζ) − ζ_{0} + \frac{1}{ζ̄_{0}} = \frac{(ζ̄_{0}ζ − R^{2})ζ}{ζ̄_{0}ζ − 1} = −ζ̄_{0}ζ · \frac{ζ − ζ_{0}}{1 − ζ̄_{0}ζ} ,

we have |F_{ζ_{0}}(ζ) − ζ_{0} + ζ̄_{0}^{−1}| = R for |ζ| = 1.

Note that the critical points ζ^{±} = (1 ± i\sqrt{R^{2} − 1})/ζ̄_{0} of the rational function F_{ζ_{0}} are contained in the unit circle. Therefore, the image of the unit circle under F_{ζ_{0}} is folded at the critical values F (ζ^{±}) = (1 ± 2i\sqrt{R^{2} − 1})/ζ̄_{0}. Since F (±e^{iα}) = −ζ̄_{0}^{−1}, we now have the required assertion. □

We denote by ℳ the set of meromorphic functions F in Δ with F (ζ) = ζ +b_{0} +O(ζ^{−1}) as ζ → ∞. Also, we denote by 𝒜 the set of analytic functions f in 𝔻 with f(z) = z+O(z^{2}) as z → 0. In the theory of univalent functions, the main object is the set 𝒮 of univalent functions in 𝒜 . For f ∈ 𝒮 , the function F (ζ) = 1/f(1/ζ) belongs to Σ and satisfies F (ζ) ≠ 0 for ζ ∈ Δ. Note then that F′(ζ) = f′(1/ζ)(ζ^{2}f(1/ζ)^{2}) = z^{2}f′(z)/f(z)^{2}, where z = 1/ζ. Therefore, Theorem 1 yields the following.

Corollary 5. For f ∈ 𝒮 , the inequality

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

holds for z ∈ 𝔻. Equality holds at z_{0} = ρe^{iα} ∈ 𝔻 \ {0} precisely when

f(z) = \frac{z}{1 − cz − e^{2iα}z^{2} \frac{1−ρ^{2}}{1−z̄_{0}z}}

for some c ∈ C_{1/z_{0}}. Here, C_{ζ_{0}} is the circular arc given in Lemma 4.

Of course, inequality (2) is a necessary condition for F to be univalent. As a sufficient condition, we have the following criterion due to Aksent’ev.

Theorem C (Aksent’ev [1]). If F ∈ ℳ satisfies the inequality

(4)  |F′(ζ) − 1| ≤ 1, |ζ| > 1,

then F is univalent in Δ.

Note that if F is meromorphically convex, namely, if Re (1 + ζF′′(ζ)/F′(ζ)) > 0 in |ζ| > 1, then F satisfies (4) (see [7, Cor. 2.5]). Krzyż [4] further showed that if |F′(ζ)−1| ≤ k < 1, then F extends to a k-quasiconformal mapping of ℂ̂.

In relation with these results, for k > 0 Ponnusamy and others studied the class 𝒰 (k) consisting of functions f ∈ 𝒜 satisfying

|\frac{z^{2}f′(z)}{f(z)^{2}} − 1| ≤ k, |z| < 1

(see, for instance, [6]). Aksent’ev’s theorem implies that 𝒰 (1) ⊂ 𝒮 .

For F ∈ ℳ and R > 1, we define F_{R} ∈ ℳ by F_{R}(ζ) = F (Rζ)/R. Since F′_{R}(ζ) = F′(Rζ), Theorem 1 yields

|F′_{R}(ζ) − 1| ≤ \frac{1}{R^{2}|ζ|^{2} − 1} < \frac{1}{R^{2} − 1}, |ζ| > 1.

In particular, for R ≥ \sqrt{1 + 1/k}, we have |F′_{R}(ζ) − 1| ≤ k, |ζ| > 1. We single out the following from this observation.

Corollary 6. Let f ∈ 𝒮 and k > 0. Then for 0 < r ≤ 1/\sqrt{1 + 1/k}, the function f_{r}(z) = rf(z/r) belongs to the class 𝒰 (k).

References

1. L. A. Aksent’ev, Sufficient conditions for univalence of regular functions (Russian), Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 3 (4), 3–7.

2. P. L. Duren, Univalent Functions, Springer-Verlag, 1983.

3. G. M. Goluzin, Geometric theory of functions of a complex variable, American Mathematical Society, Providence, R.I., 1969, Translations of Mathematical Monographs, Vol. 26.

4. J. G. Krzyż, Convolution and quasiconformal extension, Comment. Math. Helv. 51 (1976), 99–104.

5. K. Löwner, Über Extremumsätze bei der konformen Abbildung des Äußeren des Einheitskreises, Math. Z. 3 (1919), 65–77.

6. M. Obradović and S. Ponnusamy, New criteria and distortion theorems for univalent functions, Complex Variables Theory Appl. 44 (2001), 173–191.

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

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp