A COEFFICIENT INEQUALITY FOR BLOCH FUNCTIONS WITH

APPLICATIONS TO UNIFORMLY LOCALLY UNIVALENT FUNCTIONS

TOSHIYUKI SUGAWA AND TAKAO TERADA

Abstract. We give a Fekete-Szegö type inequality for a Bloch function with Bloch seminorm ≤ 1. As an application of it, we derive a sharp coefficient inequality for a_{3} for a uniformly locally univalent function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … on the unit disk with pre-Schwarzian norm ≤ λ for a given λ > 0.

1. Introduction

Let 𝒮 be the class of univalent (analytic) functions f on the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} normalized by f(0) = 1 and f′(0) = 1. Thus a function f in 𝒮 can be expanded in the form

f(z) = z + a_{2}z^{2} + a_{3}z^{3} + …, |z| < 1.

Bieberbach proved the inequality |a_{2}| ≤ 2 and conjectured that |a_{n}| ≤ n holds for every n in 1916. After the proof of |a_{3}| ≤ 3 by Löwner in 1923, Fekete and Szegö [3] surprised mathematicians by showing that the complicated inequality

|a_{3} − μa_{2}^{2}| ≤ 1 + 2 exp (\frac{−2μ}{1 − μ})

holds and is best possible for each 0 ≤ μ ≤ 1. We remark that a_{3}−a_{2}^{2} equals S_{f}(0)/6, where S_{f} is the Schwarzian derivative of f : S_{f} = (f′′/f′)′ − (f′′/f′)^{2}/2. The above inequality suggests that the shape of the coefficient region {(a_{2}, a_{3}) ∈ ℂ^{2} : ∃f ∈ 𝒮 such that f(z) = z+a_{2}z^{2}+a_{3}z^{3}+… } is quite complicated. Note that this coefficient region was thoroughly investigated by Schaeffer and Spencer [6].

In general, given a class ℱ of normalized analytic functions on the unit disk 𝔻 and a real (or, more generally, a complex) number μ, the Fekete-Szegö problem asks to find the best possible constant C(μ) so that |a_{3} − μa_{2}^{2}| ≤ C(μ) for every function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … in ℱ . Many papers have been devoted to this problem (see, for instance, [2] and references therein).

A function F on 𝔻 is called a Bloch function if the Bloch seminorm

∥F ∥_{ℬ} = sup_{z∈𝔻}(1 − |z|^{2})|F′(z)|

Date: October 9, 2006, File:  sugawa-terada06.tex.

Key words and phrases. uniformly locally univalent function, Bloch function, pre-Schwarzian derivative, Fekete-Szegö inequality.

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

is finite. We denote by ℬ the complex Banach space consisting of Bloch functions F on 𝔻 normalized by F (0) = 0 and set ℬ_{1} = {F ∈ ℬ : ∥F ∥_{ℬ} ≤ 1}. Our first principal result is stated as follows.

Theorem 1. Let μ ∈ ℂ. Then the sharp inequality

|b_{2} + μb_{1}^{2}| ≤ \begin{cases}\frac{1 + 3\sqrt{3}|μ|^{3} + (1 + 3|μ|^{2})^{3/2}}{6\sqrt{3}|μ|^{2}} & (|μ| > \frac{4}{3\sqrt{3}}) \\ \frac{3\sqrt{3}}{4} & (|μ| ≤ \frac{4}{3\sqrt{3}}) \end{cases}

holds for every function F (z) = b_{1}z + b_{2}z^{2} + … in ℬ_{1}.

The inequality in Theorem 1 can be regarded as a variant of the Fekete-Szegö inequality for ℬ_{1}.

An analytic function f on 𝔻 is called uniformly locally univalent if there is a constant ρ = ρ(f) such that f is univalent in each hyperbolic disk of radius ρ. It is known that f is uniformly locally univalent if and only if the norm

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

is finite, where T_{f} = f′′/f′ is the pre-Schwarzian derivative of f. It is also known that f is (globally) univalent if ∥T_{f}∥_{𝔻} ≤ 1 and, conversely, ∥T_{f}∥_{𝔻} ≤ 6 holds if f is univalent. We denote by 𝒰 the class of uniformly locally univalent functions f on 𝔻 normalized by f(0) = 0 and f′(0) = 1. Let 𝒰 (λ) be the subclass of 𝒰 consisting of those functions f satisfying ∥T_{f}∥_{𝔻} ≤ λ.

In [4] Y. C. Kim and the first author observed various properties of uniformly locally univalent functions. They obtained, among others, the asymptotic estimate a_{n} = O(n^{α}) for every function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … in 𝒰 (λ) and every number α with α < (\sqrt{1 + λ^{2}} − 3)/2. However, they did not have a sharp coefficient inequality except for the trivial one: |a_{2}| ≤ λ/2. We apply Theorem 1 to obtain the following result.

Theorem 2. Let λ > 0. Then the sharp inequality

|a_{3}| ≤ \begin{cases}\frac{8 + 3\sqrt{3}λ^{3} + (4 + 3λ^{2})^{3/2}}{36\sqrt{3}λ} & (λ > \frac{8}{3\sqrt{3}}) \\ \frac{\sqrt{3}}{4}λ & (λ ≤ \frac{8}{3\sqrt{3}}) \end{cases}

holds for every function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … in 𝒰 (λ).

2. Proof of Theorems 1 and 2

For a positive integer n, we consider the set

B_{n} = {(b_{1}, …, b_{n}) ∈ ℂ^{n} : ∃F ∈ ℬ_{1} such that F (z) = b_{1}z + ⋯ + b_{n}z^{n} + … },

which is sometimes called the coefficient region of ℬ_{1} with order n. Bonk studied in his dissertation [1] the coefficient regions B_{n} and observed that they are closed convex sets with non-empty interior. It is an easy exercise to show that B_{1} = {|b_{1}| ≤ 1}. One of Bonk’s main contributions was to give a description of B_{2}. To state his result, we need to introduce auxiliary functions. Let

P (x) = \frac{3\sqrt{3}}{2}x(1 − x^{2}).

Then the function P (x) increases from 0 to 1 when x moves from 0 to 1/\sqrt{3}. Therefore, we can take a branch Q of P^{−1} on the interval [0, 1] so that Q : [0, 1] → [0, 1/\sqrt{3}] is homeomorphic. Note that the relation

(2.1)  P (Q(t)) = \frac{3\sqrt{3}}{2}Q(t)(1 − Q(t)^{2}) = t

holds for t ∈ [0, 1]. We are now ready to state Bonk’s theorem.

Theorem A (Bonk [1, Satz 3.2.1]).

B_{2} = \{(b_{1}, b_{2}) ∈ ℂ^{2} : |b_{1}| ≤ 1 and |b_{2}| ≤ \frac{3\sqrt{3}}{4}(1 − 3Q(|b_{1}|)^{2})(1 − Q(|b_{1}|)^{2})\} .

In particular, we have the sharp bound |b_{2}| ≤ 3\sqrt{3}/4 for functions F (z) = b_{1}z+b_{2}z^{2}+… in ℬ_{1}. With this information about B_{2}, we prove Theorem 1.

Proof of Theorem 1. Let C(μ) be the best possible constant C such that |b_{2} + μb_{1}^{2}| ≤ C holds for every function F (z) = b_{1}z +b_{2}z^{2} +… in ℬ_{1}, where μ is a fixed complex number. Then, by definition of the coefficient region, we have

C(μ) = sup_{(b_{1},b_{2})∈B_{2}} |b_{2} + μb_{1}|.

For (b_{1}, b_{2}) ∈ B_{2}, by Theorem A,

(2.2)  |b_{2} + μb_{1}^{2}| ≤ |b_{2}| + |μ||b_{1}|^{2}

≤ \frac{3\sqrt{3}}{4}(1 − 3Q(|b_{1}|)^{2})(1 − Q(|b_{1}|)^{2}) + |μ||b_{1}|^{2} = M(|b_{1}|), (2.3)

where

M(t) = \frac{3\sqrt{3}}{4}(1 − 3Q(t)^{2})(1 − Q(t)^{2}) + |μ|t^{2}.

We note here that we can choose (b_{1}, b_{2}) ∈ B_{2} so that equality holds at both (2.2) and (2.3). Since |b_{1}| can take any value in [0, 1], we obtain

(2.4)  C(μ) = max_{0≤t≤1} M(t).

We have thus to compute the value of the maximum of M(t) over 0 ≤ t ≤ 1. Since P′(Q(t))Q′(t) = 1, we obtain the relation

Q′(t) = \frac{2}{3\sqrt{3}(1 − 3Q(t)^{2})}.

Therefore, by substituting the last relation and (2.1), we get

M′(t) = −3\sqrt{3}(2 − 3Q(t)^{2})Q(t)Q′(t) + 2|μ|t

= −\frac{2Q(t)(2 − 3Q(t)^{2})}{1 − 3Q(t)^{2}} + 3\sqrt{3}|μ|Q(t)(1 − Q(t)^{2})

= \frac{Q(t)}{1 − 3Q(t)^{2}} \{2(3Q(t)^{2} − 2) + 3\sqrt{3}|μ|(Q(t)^{2} − 1)(3Q(t)^{2} − 1)\} .

Solving the quadratic equation 2(3x−2)+3\sqrt{3}|μ|(x−1)(3x−1) = 0, we have the solutions x = (2\sqrt{3}|μ| − 1 ± \sqrt{1 + 3|μ|^{2}})/(3\sqrt{3}|μ|). Because (2\sqrt{3}|μ| − 1 + \sqrt{1 + 3|μ|^{2}})/(3\sqrt{3}|μ|) ≥ 2/3 > 1/3, if the derivative M′(t) has a zero t_{0} in the interval (0, 1) it must satisfy the relation

Q(t_{0})^{2} = \frac{2\sqrt{3}|μ| − 1 − \sqrt{1 + 3|μ|^{2}}}{3\sqrt{3}|μ|} .

We now set

R(s) = \frac{2\sqrt{3}s − 1 − \sqrt{1 + 3s^{2}}}{3\sqrt{3}s} , s > 0.

Since

R′(s) = \frac{1 + \sqrt{1 + 3s^{2}}}{3s\sqrt{3(1 + 3s^{2})}} > 0,

the function R(s) is increasing in s > 0. Note that R(\frac{4}{3\sqrt{3}}) = 0 and lim_{s→+∞} R(s) = \frac{1}{3}. Therefore, the equation Q(t)^{2} = R(|μ|) has a solution t = t_{0} in the interval (0, 1) precisely when \frac{4}{3\sqrt{3}} < |μ|.

First we consider the case when |μ| ≤ \frac{4}{3\sqrt{3}}. In this case, M′(t) < 0 in 0 < t < 1 and hence M(|b_{1}|) takes its maximum as |b_{1}| = 0. Therefore, we obtain C(μ) = M(0) = 3\sqrt{3}/4 by (2.4).

Secondly, we assume that |μ| > \frac{4}{3\sqrt{3}}. Then, as was seen above, there is a unique point t_{0} ∈ (0, 1) such that Q(t_{0})^{2} = R(|μ|). Since M′(t) > 0 for 0 < t < t_{0} and M′(t) < 0 for t_{0} < t < 1, the function M(t) takes its maximum at t = t_{0}. Thus, C(μ) = M(t_{0}) by (2.4). Let us now compute the value of M(t_{0}). In view of the relation t_{0} = P (Q(t_{0})) = P (\sqrt{R(|μ|)}), we have the expression

M(t_{0}) = \frac{3\sqrt{3}}{4}(1 − 3Q(t_{0})^{2})(1 − Q(t_{0})^{2}) + |μ|t_{0}^{2}

= \frac{3\sqrt{3}}{4}(1 − 3R(|μ|))(1 − R(|μ|)) + |μ|P (\sqrt{R(|μ|)})^{2}

= \frac{1 + 3\sqrt{3}|μ|^{3} + (1 + 3|μ|^{2})^{3/2}}{6\sqrt{3}|μ|^{2}} .

Thus, the assertion of Theorem 1 has been confirmed. □

Proof of Theorem 2. For a function f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … in 𝒰 (λ), we set F = λ^{−1} log f′. Then, ∥F ∥_{ℬ} = λ^{−1}∥T_{f}∥_{𝔻} ≤ 1 and thus F ∈ ℬ_{1}. We expand F in a power series: F (z) = b_{1}z + b_{2}z^{2} + …. A comparison of the Taylor coefficients of the both sides of f′ = e^{λF} yields the relations

2a_{2} = λb_{1} and 3a_{3} = λ (b_{2} + \frac{λ}{2}b_{1}^{2}) .

Thus, the maximum of |a_{3}| for f ∈ 𝒰 (λ) is given as λC(λ/2)/3. Theorem 1 now yields the required assertion. □

Under the same circumstances as in the above proof, we further obtain the expression

a_{3} − μa_{2}^{2} = \frac{λ}{3} [b_{2} + \frac{λ}{4}(2 − 3μ)b_{1}^{2}] .

Hence, as an immediate consequence of Theorem 1, we also have the Fekete-Szegö inequality for the class 𝒰 (λ).

Theorem 3. Let a fimctopm f(z) = z + a_{2}z^{2} + a_{3}z^{3} + … belong to 𝒰 (λ) for a λ > 0. Then the sharp inequality

|a_{3} − μa_{2}^{2}| ≤ \frac{λ}{3}C(\frac{(2−3μ)λ}{4})

= \begin{cases}\frac{64 + 3\sqrt{3}λ^{3}|2 − 3μ|^{3} + (16 + 3λ^{2}|2 − 3μ|^{2})^{3/2}}{72\sqrt{3}λ|2 − 3μ|^{2}} & (λ|2 − 3μ| > \frac{16}{3\sqrt{3}}) \\ \frac{\sqrt{3}}{4}λ & (λ|2 − 3μ| ≤ \frac{16}{3\sqrt{3}}) \end{cases}

holds for each μ ∈ ℂ.

Since S_{f}(0) = 6(a_{3} − a_{2}^{2}), we obtain the following corollary.

Corollary 4. For f ∈ 𝒰 (λ), λ > 0, the sharp inequality

|S_{f}(0)| ≤ 2λC(−λ/4) = \begin{cases}\frac{64 + 3\sqrt{3}λ^{3} + (16 + 3λ^{2})^{3/2}}{12\sqrt{3}λ} & (λ > \frac{16}{3\sqrt{3}}) \\ \frac{3\sqrt{3}}{2}λ & (λ ≤ \frac{16}{3\sqrt{3}}) \end{cases}

holds.

3. Extremal functions

We end the paper with a remark on functions extremal in 𝒰 (λ). First we observe extremal functions for the coefficient functional |b_{2} + μb_{1}^{2}| in ℬ_{1}. It is clear that such an extremal function F (z) = b_{1}z + b_{2}z^{2} + … has to satisfy the condition (b_{1}, b_{2}) ∈ ∂B_{2}, in other words, either

(i) |b_{1}| = 1 and b_{2} = 0, or (ii) |b_{1}| < 1 and |b_{2}| = \frac{3\sqrt{3}}{4}(1 − 3Q(|b_{1}|)^{2})(1 − Q(|b_{1}|)^{2}).

In case (i), an extremal function is given by F (z) = b_{1}z. In case (ii), setting t_{0} = P (|b_{1}|), we define F by

F (z) = \frac{3\sqrt{3}ε}{4} \{(\frac{z + z_{0}}{1 + \overline{z_{0}}z} )^{2} − z_{0}^{2}\} ,

where ε ∈ ∂𝔻 and z_{0} ∈ 𝔻 are chosen so that arg ε = arg b_{2}, |z_{0}| = Q(|b_{1}|), and arg z_{0} = arg b_{1} − arg b_{2}. Then, it is checked that ∥F ∥_{ℬ} = 1, F′(0) = ε(z_{0}/|z_{0}|)P (|z_{0}|) = b_{1} and F′′(0)/2 = ε\frac{3\sqrt{3}}{4}(1 − 3|z_{0}|^{2})(1 − |z_{0}|^{2}) = b_{2}. Therefore, F (z) = b_{1}z + b_{2}z^{2} + ….

As for uniqueness of extremal functions, at least, we have the following.

Lemma 5. Let (b_{1}, b_{2}) ∈ ∂B_{2}. If |b_{1}| = 1, then there are infinitely many functions F ∈ ℬ_{1} such that F (z) = b_{1}z+O(z^{3}). If b_{1} = 0 then a function F ∈ ℬ_{1} with F (z) = b_{1}z+b_{2}z^{2}+… necessarily has the form F (z) = b_{2}z^{2}.

Proof. We may first assume that b_{1} = 1. Let ω be an analytic map of 𝔻 into itself with ω(0) = ω′(0) = 0. Then, consider the function

F (z) = ∫_{0}^{z} \frac{dζ}{1 − ω(ζ)} = ∫_{0}^{1} \frac{zdt}{1 − ω(tz)}.

Then F is analytic on 𝔻 and satisfies F (0) = 0 and F′(0) = 1. On the other hand, since |ω(z)| ≤ |z|^{2}, we have

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

for |z| < 1 with equality for z = 0. Thus, we see that ∥F ∥_{ℬ} = 1. In this way, we can construct a plenty of such functions.

Next we assume that b_{1} = 0 and |b_{2}| = 3\sqrt{3}/4. Let F be a function in ℬ_{1} such that F (z) = b_{2}z^{2} + c_{3}z^{3} + c_{4}z^{4} + …. Then,

F′(z) = 2b_{2}z + 3c_{3}z^{2} + ⋯ = F′_{0}(z)(1 + h(z)),

where F_{0}(z) = b_{2}z^{2} and h is analytic on 𝔻 with h(0) = 0. In particular, we have

(1 − |z|^{2})|F′(z)| = \frac{3\sqrt{3}}{2}|z|(1 − |z|^{2})|1 + h(z)| = |1 + h(z)| ≤ 1

for |z| = 1/\sqrt{3}, by the assumption ∥F ∥_{ℬ} ≤ 1. By the maximum modulus principle, this forces h to be identically 0. Thus, the proof is complete. □

As conseqences of the last lemma together with the proof of Theorem 1, we can deduce some information about extremal functions in ℬ_{1} and 𝒰 (λ).

Theorem 6. Let μ ∈ ℂ satisfy |μ| ≤ \frac{4}{3\sqrt{3}}. Then an extremal function F_{0} for the coefficient functional |b_{2} + μb_{1}^{2}| for functions F (z) = b_{1}z + b_{2}z^{2} + … in ℬ_{1} must have the form F_{0}(z) = ε\frac{3\sqrt{3}}{4}z^{2} for a complex constant ε with |ε| = 1.

We recall the definition of the error function:

Erf(z) = ∫_{0}^{z} e^{−ζ^{2}}dζ.

Then extremal functions in 𝒰 (λ) can be expressed in terms of the error function for a small λ.

Theorem 7. Let 0 < λ ≤ \frac{8}{3\sqrt{3}}. Suppose that a function f ∈ 𝒰 (λ) maximizes the functional |a_{3}| within 𝒰 (λ). Then f has to be represented by

f(z) = \frac{Erf(αz)}{α}

for a complex constant α with |α|^{2} = 3\sqrt{3}λ/4.

Kreyszig and Todd [5] obtained the radius ρ of univalence of the error function up to 7 decimal places by using large-scale computers. According to their observations, the radius ρ is given by ρ = \sqrt{(θ + π/2)/ sin 2θ}, where θ ∈ (0, π/2) is determined by the equation

Im Erf (\sqrt{\frac{θ + π/2}{sin 2θ}}e^{iθ}) = 0.

Using Mathematica 5.2, we obtained numerically

ρ = 1.57483758917543224805 ….

Especially, we admit that their computation was correct. Since α in Theorem 7 satisfies |α| ≤ \sqrt{2} = 1.414 …, the extremal function f(z) = Erf(αz)/α is univalent in the unit disk for such an α.

References

1. M. Bonk, Extremalprobleme bei Bloch-funktionen, Dissertation, TU Braunschweig (1988).

2. J. H. Choi, Y. C. Kim, and T. Sugawa, A general approach to the Fekete-Szegö problem, Preprint (2003).

3. M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85–89.

4. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math. 32 (2002), 179–200.

5. E. Kreyszig and J. Todd, The radius of univalence of the error function, Numer. Math. 1 (1959), 78–89.

6. A. C. Schaeffer and D. C. Spencer, Coefficient regions for schlicht functions, Amer. Math. Soc. Colloq. Publ., vol. 35, 1950.

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

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

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

  E-mail address: teradat@hiroshima-u.ac.jp