Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 419–432

NECESSARY AND SUFFICIENT CONDITIONS FOR THE BERNSTEIN INEQUALITY

Kiwon Kim

Pusan Women’s University, Department of Mathematics Goaebop-dong, Buk-gu, Pusan, 616-060, Korea; kwkim@lotus.pwu.ac.kr

Abstract. The closure E of a k -quasidisk with 0 ≤ k < 1 satisfies the Bernstein inequality

∥p′∥_{E} ≤ a \frac{n^{1+k}}{tr(E)} ∥p∥_{E},

where a = 2^{−k}e [AGH]. In this paper, we extend the above result to the case of the closure of a c-John disk with an absolute constant a and a constant k, 0 ≤ k < 1, which depends only on a John constant c. We also give a characterization of a bounded continuum which satisfies the Bernstein inequality in terms of a normalized exterior conformal mapping.

1. Introduction

Let B denote the open unit disk in the complex plane C and D a bounded Jordan domain in C. For the purpose of this paper we say that D is an open k - quasidisk, 0 ≤ k < 1, if one and hence each conformal mapping g: \overline{C} \ \overline{B} → \overline{C} \ \overline{D} can be extended to a K -quasiconformal mapping of the extended complex plane \overline{C} where K = (1 + k)/(1 − k). A continuum E ⊂ C is said to be a closed k - quasidisk, if E = \overline{D} where D is as above. Finally a bounded continuum E whose complement in \overline{C} is connected is said to be a closed 1-quasidisk.

A bounded simply connected domain D ⊂ C is said to be a c-John disk if there exist a point z_{0} ∈ D and a constant c ≥ 1 such that each point z_{1} ∈ D can be joined to z_{0} by an arc γ in D satisfying

ℓ(γ(z_{1}, z)) ≤ c d(z, ∂D)

for each z ∈ γ. We call z_{0} a John center, c a John constant and γ a c-John arc. Thus the closure of a John disk is a closed 1-quasidisk. A quasidisk is a John disk. But the converse is not true since a John disk need not even be a Jordan domain.

Suppose that E is a closed quasidisk. If g: \overline{C} \ \overline{B} → \overline{C} \ E is conformal with g(∞) = ∞, then

g(w) = a_{−1}w + ∑_{n=0}^{∞} a_{n}w^{−n}, |w| > 1,

1991 Mathematics Subject Classification: Primary 30C65.

and the number |a_{−1}| is called the transfinite diameter of E, denoted by tr(E). Let P_{n} denote the class of polynomials p of degree at most n and

∥p∥_{E} = max{|p(z)| : z ∈ E}.

At first, we establish sufficient conditions for the Bernstein inequality. The following beautiful inequality is due to Bernstein [C].

Lemma 1.1. Bernstein inequality: If E is the closure of a euclidean disk, then

∥p′∥_{E} ≤ \frac{n}{tr(E)}∥p∥_{E}

for all p ∈ P_{n}.

In [AGH], Anderson, Gehring and Hinkkanen extended the above result to the case where E is a closed k -quasidisk with 0 ≤ k ≤ 1 as follows:

Lemma 1.2. If E is a closed k -quasidisk with 0 ≤ k ≤ 1, then for each p ∈ P_{n}

∥p′∥_{E} ≤ c_{1} \frac{n^{1+k}}{tr(E)}∥p∥_{E},

where c_{1} = 2^{−k}e.

Since the closure E of a John disk is a closed 1-quasidisk, by Lemma 1.2 E satisfies the inequality

∥p′∥_{E} ≤ \frac{e}{2} \frac{n^{2}}{tr(E)}∥p∥_{E},

a result originally first proved by Pommerenke [P2].

In this paper, we extend Lemma 1.2 with k ∈ [0, 1) to the case where E is the closure of a John disk D, by using the quasidisk property (Theorem 2.1) for a John disk.

Theorem 1.3. Suppose that D is a c-John disk with a John center z_{0} , that E is the closure of D and that p is a polynomial in z of degree n. Then

sup_{E} |p′(z)| ≤ a\frac{n^{b}}{d(z_{0}, ∂D)} sup_{E} |p(z)|,

where a is an absolute constant and b is a constant in [1, 2) which depends only on c.

In Theorem 1.3 we see that if E is the closure of a c-John disk with tr(E) = 1, then the Bernstein inequality

(1.4)  sup_{E} |p′(z)| ≤ an^{b} sup_{E} |p(z)|

holds for each polynomial p in z of degree n, where a and b, 1 ≤ b < 2, depend only on c.

We see in Remark 2.10 that there exists a set for which (1.4) holds with a = b = 1, while D = int (E) is not connected and hence not a John disk.

Remark 1.5. Given bounded continuum E, let G^{∗} be the component of \overline{C} \ E which contains ∞. Let F = \overline{C} \ G^{∗} and let g: \overline{C} \ \overline{B} → G^{∗} be a conformal mapping with g(∞) = ∞. Then E ⊂ F and from the proof of Lemma 1.2 in [AGH], we see that if 1 ≤ b < 2 and if

(1.6)  |g′(w)| ≥ m(1 − |w|^{−2})^{b−1}

for 1 < |w| < \sqrt{2} + 1, then

sup_{F} |p′(z)| ≤ an^{b} sup_{F} |p(z)|

for each p ∈ P_{n} where a depends only on b and tr(F).

Thus (1.6) is a sufficient condition for the Bernstein inequality to hold on a bounded continuum E in C with connected complement which contains ∞.

One of the main purposes of this paper is to show that (1.6) gives also a necessary condition for the Bernstein inequality on the E (Theorem 1.9).

Lemma 1.7. Suppose that E is a bounded continuum in C for which (1.4) holds for some constants a and b, 1 ≤ b < 2 and for each polynomial p in z of degree n. Then (1.4) holds with E replaced by F = \overline{C} \ D^{∗} where D^{∗} is the component of \overline{C} \ E which contains ∞.

Proof. Since D^{∗} contains ∞, F is a bounded continuum. Fix z_{0} ∈ F \ E and let G be a component of F \ E = (\overline{C} \ E) \ D^{∗} which contains z_{0} . If p is a polynomial of degree n, then p′ is analytic in G and by the maximum principle

|p′(z_{0})| ≤ sup_{z∈∂G} |p′(z)| ≤ sup_{z∈E} |p′(z)| ≤ an^{b} sup_{z∈E} |p(z)| ≤ an^{b} sup_{z∈F} |p(z)|.

Therefore

sup_{F} |p′(z)| ≤ an^{b} sup_{F} |p(z)|.

Also we may assume without loss of generality that tr(E) = 1 by performing a preliminary similarity mapping. Thus by Lemma 1.7 we assume E satisfies the following hypothesis.

Hypothesis 1.8. E is a bounded continuum in C with connected complement D^{∗} and tr(E) = 1, a and b are constants such that 1 ≤ b < 2 and such that (1.4) holds for each polynomial p in z of degree n, and

g(w) = w + ∑_{n=0}^{∞} a_{n}w^{−n}, |w| > 1

maps B^{∗} = \overline{C} \ \overline{B} conformally onto D^{∗} so that g(∞) = ∞.

Theorem 1.9. If E satisfies Hypothesis 1.8, then for each constant c, b < c < 2,

|g′(w)| ≥ m(1 − |w|^{−2})^{c−1}

for 1 < |w| < \sqrt{2} + 1, where m is a constant which depends only on a, b, c.

Therefore, by Remark 1.5 and by Theorem 1.9 we obtain a characterization of any bounded continuum which satisfies Hypothesis 1.8 in terms of the normalized exterior mapping g.

2. The proof of Theorem 1.3

Gehring and Osgood show in [GO] that a domain D in C is uniform if and only if it is quasiconformally decomposable, i.e., for each z_{1}, z_{2} ∈ D there exists a K -quasidisk G_{0} in D such that z_{1}, z_{2} ∈ \overline{G_{0}} where K = K(D). We give a geometric characterization of John disks which is the analogue of the above property of uniform domains.

We say that a domain D in C has the quasidisk property if for some fixed point z_{0} = z_{0}(D) ∈ D and for each z_{1} ∈ D, there exists a K -quasidisk G_{1} in D with z_{0}, z_{1} ∈ \overline{G_{1}} , where K = K(D).

Theorem 2.1. A bounded Jordan domain D in C is a c-John disk if and only if it has the quasidisk property.

The proof of Theorem 2.1 depends on three lemmas.

Lemma 2.2 ([GHM, Theorem 4.1]). If D is a c-John disk with a John center z_{0} and if γ is a hyperbolic geodesic which joins z_{1} to z_{0} for z_{1} ∈ D, then γ is a b-John arc for some constant b which depends only on c.

Lemma 2.3 ([GH] and [J]). Suppose that D is a Jordan domain in C. If γ is a hyperbolic geodesic in D and if α is any curve which joins the end points of γ in D, then

ℓ(γ) ≤ kℓ(α),

where k is an absolute constant, 4.5 ≤ k ≤ 17.5.

Lemma 2.4. Let D be a c-John disk with a John center z_{0} and let γ be a hyperbolic geodesic with z_{0} as one of its endpoints. If z_{1}, z_{2} ∈ γ and if z_{1} separates z_{0} and z_{2} , then

ℓ(γ(z_{1}, z_{2})) ≤ b min(|z_{1} − z_{2}|, d(z_{1}, ∂D))

where b is a constant which depends only on c.

Proof of Lemma 2.4. Fix z_{1}, z_{2} ∈ γ. By Lemma 2.2,
(2.5)  ℓ(γ(z_{1}, z_{2})) ≤ b_{1} d(z_{1}, ∂D)

for some constant b_{1} which depends only on c.

If |z_{1} − z_{2}| ≥ d(z_{1}, ∂D), then by (2.5)

(2.6)  ℓ(γ(z_{1}, z_{2})) ≤ b_{1} |z_{1} − z_{2}|.

If |z_{1} − z_{2}| < d(z_{1}, ∂D), then the segment [z_{1}, z_{2}] joining z_{1} and z_{2} lies in D and

(2.7)  ℓ(γ(z_{1}, z_{2})) ≤ kℓ([z_{1}, z_{2}]) = k|z_{1} − z_{2}|,

by Lemma 2.3 for an absolute constant k > 0. Hence (2.5), (2.6) and (2.7) complete the proof of Lemma 2.4 with b = max(b_{1}, k).

Proof of Theorem 2.1. Suppose that a bounded Jordan domain D in C is a c-John disk with a John center z_{0} . Fix z_{1} ∈ D and let γ be the hyperbolic geodesic joining z_{0} and z_{1} in D. Fix w_{1}, w_{2} ∈ γ labeled so that w_{1} separates z_{0} and w_{2} in γ. Then by Lemma 2.4,

ℓ(γ(w_{1}, w_{2})) ≤ b|w_{1} − w_{2}|

where b is a constant which depends only on c. Next if z ∈ γ, then z separates z_{0} and z_{1} in γ and by Lemma 2.4

min_{j=0,1} ℓ(γ(z_{j}, z)) ≤ ℓ(γ(z, z_{1})) ≤ bd(z, ∂D).

Thus γ satisfies conditions in (4.1) of [GO] with a_{1} = b_{1} = b and the construction given on [GO, pp. 67–68] yields a K -quasidisk G_{1} with desired properties, where K = K(a_{1}, b_{1}) = K(c).

Conversely, we assume that there exist a point z_{0} ∈ D and a constant K such that for each z_{1} ∈ D, there is a K -quasidisk G_{1} in D with z_{0}, z_{1} ∈ \overline{G_{1}} . Fix z_{1} ∈ D, choose a quasidisk G_{1} in D corresponding to z_{1} and let γ be the hyperbolic geodesic joining z_{0} and z_{1} in G_{1} . Then for all z ∈ γ we have a constant a = a(K) ≥ 1 such that

(2.8)  ℓ(γ(z, z_{1})) ≤ a|z − z_{1}|

and

(2.9)  min_{j=0,1} ℓ(γ(z_{j}, z)) ≤ ad(z, ∂G_{1}) ≤ ad(z, ∂D)

[GO, Corollary 4]. Next let

\frac{b = dia (D)}{d(z_{0}, ∂D) < ∞}

and let c = 2a^{2}b. We will show that

ℓ(γ(z, z_{1})) ≤ cd(z, ∂D)

for all z ∈ γ and hence that D is a c-John disk. We consider two cases.

Suppose first that

|z − z_{0}| ≤ \frac{1}{2}d(z_{0}, ∂D).

Then

d(z, ∂D) ≥ d(z_{0}, ∂D) − |z − z_{0}| ≥ \frac{1}{2}rd(z_{0}, ∂D)

and hence by (2.8)

\begin{aligned}ℓ(γ(z, z_{1})) & ≤ a|z − z_{1}| ≤ a dia (D) = ab d(z_{0}, ∂D) \\ & ≤ 2ab d(z, ∂D) ≤ cd(z, ∂D). \end{aligned}

Suppose next that

|z − z_{0}| ≥ \frac{1}{2}d(z_{0}, ∂D).

If ℓ(γ(z_{0}, z)) ≤ ℓ(γ(z, z_{1})), then as above and by (2.9)

\begin{aligned}ℓ(γ(z, z_{1})) & ≤ adia (D) ≤ abd(z_{0}, ∂D) ≤ 2ab|z − z_{0}| \\ & ≤ 2abℓ(γ(z, z_{0})) ≤ 2a^{2}bd(z, ∂D) = cd(z, ∂D). \end{aligned}

If ℓ(γ(z_{0}, z)) ≥ ℓ(γ(z, z_{1})), then by (2.9)

ℓ(γ(z, z_{1})) ≤ ad(z, ∂D) ≤ cd(z, ∂D).

Proof of Theorem 1.3. Let z_{0} be a John center of D and let {z_{i}} be a sequence in D which converges to a point w_{0} ∈ ∂D. Then by Theorem 2.1, for each j there exists a K -quasidisk G_{j} in D with z_{0}, z_{j} ∈ \overline{G_{j}}. Also, since \overline{G_{j}} is connected,

tr(\overline{G_{j}}) ≥ \frac{1}{4}|z_{j} − z_{0}|.

By Lemma 1.2

|p′(z_{j})| ≤ sup_{\overline{G_{j}}} |p′(z)| ≤ c_{1} \frac{n^{1+k}}{tr(\overline{G_{j}})} sup_{\overline{G_{j}}} |p(z)| ≤ 4c_{1} \frac{n^{1+k}}{|z_{j} − z_{0}|} sup_{D} |p(z)|,

where k = (K − 1)/(K + 1) ∈ [0, 1), K = K(c), and c_{1} is in Lemma 1.2. Therefore

|p′(w_{0})| ≤ lim_{j→∞} 4c_{1} \frac{n^{k+1}}{|z_{j} − z_{0}|} sup_{\overline{D}} |p(z)| ≤ a\frac{n^{b}}{|w_{0} − z_{0}|} sup_{\overline{D}} |p(z)|

≤ a\frac{n^{b}}{d(z_{0}, ∂D)} sup_{\overline{D}} |p(z)|,

where a is an absolute constant and b is a constant in [1, 2) which depends only on c. Then since |p′(z)| satisfies the maximum principle, the proof of Theorem 1.3 is complete.

Remark 2.10. The converse of Theorem 1.3 is false (i.e., a set E for which such an inequality holds need not be the closure of a John disk). For example, let E be any bounded continuum of the form

E = ⋃_{j=1}^{n} \overline{D_{j}},

where D_{j}, j = 1, …, n, are mutually disjoint euclidean disks with tr(\overline{D_{j}}) ≥ 1 and ∂D_{j} ∩ ∂D_{j+1} is a point for j = 1, 2, …, n. Then E is not the closure of a John disk because its interior is not connected. However, if p is a polynomial of degree n, then by Lemma 1.1,

sup_{E} |p′(z)| = sup_{j} sup_{\overline{D_{j}}} |p′(z)| ≤ sup_{j} \frac{n}{tr(\overline{D_{j}})} sup_{\overline{D_{j}}} |p(z)| ≤ n sup_{E} |p(z)|.

Thus E satisfies the inequality.

3. The proof of Theorem 1.9

Let {p_{n}} be the Faber polynomials for g, i.e.,

\frac{g′(w)}{g(w) − z} = ∑_{n=0}^{∞} p_{n}(z)w^{−n−1}

for z ∈ E , (see [P1]).

426 Kiwon Kim Lemma 3.1. If {p_{n}} are the Faber polynomials for g, then

∑_{n=1}^{k} \frac{1}{n}|p_{n}(z)| ≤ 5 log(k + 1).

Proof of Lemma 3.1. By the Cauchy–Schwarz inequality and [P1, p. 85],

∑_{n=1}^{k} \frac{1}{n}|p_{n}(z)| ≤ (∑_{n=1}^{k} \frac{1}{n}|p_{n}(z)|^{2})^{1/2}(∑_{n=1}^{k} \frac{1}{n})^{1/2}

≤ (4 ∑_{n=1}^{k} \frac{1}{n} + 1.248)^{1/2}(∑_{n=1}^{k} \frac{1}{n})^{1/2}

≤ (6.25 ∑_{n=1}^{k} \frac{1}{n})^{1/2}(∑_{n=1}^{k} \frac{1}{n})^{1/2} = 2.5 ∑_{n=1}^{k} \frac{1}{n}.

Then since

\begin{aligned}∑_{n=1}^{k} \frac{1}{n} & = ∑_{n=1}^{k} \frac{n + 1}{n} ∫_{n}^{n+1} \frac{1}{n + 1} dt ≤ ∑_{n=1}^{k} 2 ∫_{n}^{n+1} \frac{1}{t} dt \\ & = 2 ∫_{1}^{k+1} \frac{1}{t} dt = 2 log(k + 1), \end{aligned}

we have

∑_{n=1}^{k} \frac{1}{n}|p_{n}(z)| ≤ 5 log(k + 1).

Lemma 3.2. If 1 ≤ a < 2, then

∑_{n=1}^{∞} n^{a−1}t^{n} ≤ 4(1 − t)^{−a}

for 0 ≤ t < 1.

Proof of Lemma 3.2. Let

f(t) = \frac{1}{(1 − t)^{a}} .

Then

\frac{f^{(n)}(0)}{n!} = \frac{a(a + 1) ⋯ (a + n − 1)}{n!} = \frac{Γ(a + n)}{Γ(a)Γ(n + 1)}.

Necessary and sufficient conditions for the Bernstein inequality 427 Thus we have

(3.3)  ∑_{n=1}^{∞} \frac{Γ(a + n)}{Γ(a)Γ(n + 1)}t^{n} < \frac{1}{(1 − t)^{a}}

for 0 ≤ t < 1. Next for x > 0

Γ(x) = \sqrt{2π} x^{x−1/2}e^{−x}e^{θ(x)/12x},

where 0 < θ(x) < 1 by Stirling’s formula on [A, p. 206]. Hence

(3.4) \begin{aligned}\frac{Γ(a + n)}{Γ(n + 1)} & = \frac{(a + n)^{a+n−1/2}e^{−(a+n)}e^{θ(a+n)/12(a+n)}}{(n + 1)^{n+1−1/2}e^{−(n+1)}e^{θ(n+1)/12(n+1)}} \\ & ≥ (\frac{a − 1 + n + 1}{n + 1})^{n+1/2}(\frac{a + n}{n})^{a−1}n^{a−1}e^{−1/12(n+1)}e^{−a+1} \\ & ≥ (1 + \frac{a − 1}{n + 1})^{n+1/2}(1 + \frac{a}{n})^{a−1}n^{a−1}e^{−1−1/24} \\ & ≥ (1 + \frac{a − 1}{n + 1})^{n+1}(1 + \frac{a − 1}{n + 1})^{a−3/2}n^{a−1}e^{−25/24}. \end{aligned}

Since

1 ≤ 1 + \frac{a − 1}{n + 1} < \frac{3}{2} and −\frac{1}{2} ≤ a − \frac{3}{2} < \frac{1}{2},

we have

(1 + \frac{a − 1}{n + 1})^{n+1} ≥ 1 and (1 + \frac{a − 1}{n + 1})^{a−3/2} ≥ \sqrt{\frac{2}{3}}.

Thus by (3.4)

n^{a−1} ≤ \sqrt{\frac{3}{2}} e^{25/24} \frac{Γ(a + n)}{Γ(n + 1)} ≤ 4\frac{Γ(a + n)}{Γ(a)Γ(n + 1)},

whence by (3.3)

∑_{n=1}^{∞} n^{a−1}t^{n} ≤ 4 ∑_{n=1}^{∞} \frac{Γ(a + n)}{Γ(a)Γ(n + 1)}t^{n} ≤ 4(1 − t)^{−a}

for 0 ≤ t < 1.

Lemma 3.5. If E satisfies Hypothesis 1.8, then for each constant c, b < c < 2,

|g(w) − z| ≥ m(1 − \frac{1}{|w|})^{c}

for z ∈ E and 1 < |w| < ∞, where m is a constant which depends only on a, b, c.

Proof of Lemma 3.5. Let {p_{n}} be the Faber polynomial for g(w). Then by [P1, p. 57]

(3.6)  \begin{aligned}\frac{1}{|g(w) − z|} & = |∑_{n=1}^{∞} \frac{1}{n}p′_{n}(z)w^{−n}| \\ & ≤ ∑_{k=0}^{∞} | ∑_{n=2^{k}}^{2^{k+1}−1} \frac{1}{n}p′_{n}(z)w^{−n}| = ∑_{k=0}^{∞} |q′_{k}(z)|, \end{aligned}

where

q′_{k}(z) = ∑_{n=2^{k}}^{2^{k+1}−1} \frac{1}{n}p′_{n}(z)w^{−n}.

Thus q_{k}(z) is a polynomial of degree at most 2^{k+1} − 1. Hence by Hypothesis 1.8 we get

(3.7)  |q′_{k}(z)| ≤ a(2^{k+1} − 1)^{b} sup_{ξ∈E} |q_{k}(ξ)| ≤ a(2^{k+1})^{b} sup_{ξ∈E} |q_{k}(ξ)|.

Next by Lemma 3.1 we have

(3.8)  \begin{aligned}|q_{k}(ξ)| & = | ∑_{n=2^{k}}^{2^{k+1}−1} \frac{1}{n}p_{n}(ξ)w^{−n}| ≤ ∑_{n=2^{k}}^{2^{k+1}−1} \frac{1}{n}|p_{n}(ξ)| |w|^{−n} \\ & ≤ ( ∑_{n=1}^{2^{k+1}−1} \frac{1}{n}|p_{n}(ξ)|)|w|^{−2^{k}} ≤ 5 log (2^{k+1} − 1 + 1)|w|^{−2^{k}} \\ & = (k + 1)5 log 2|w|^{−2^{k}} ≤ 4(k + 1)|w|^{−2^{k}} \end{aligned}

for ξ ∈ E. Hence by (3.6), (3.7) and (3.8),

(3.9)  \begin{aligned}\frac{1}{|g(w) − z|} & = ∑_{k=0}^{∞} |q′_{k}(z)| ≤ ∑_{k=0}^{∞} a(2^{k+1})^{b} sup_{ξ∈E} |q_{k}(ξ)| \\ & ≤ ∑_{k=0}^{∞} a(2^{k+1})^{b}4(k + 1)|w|^{−2^{k}} \\ & = ∑_{k=0}^{∞} 2^{b+2}a(k + 1)2^{kb}|w|^{−2^{k}}. \end{aligned}

Necessary and sufficient conditions for the Bernstein inequality 429 Next let

f(x) = (x + 1)2^{(b−c)x}

for b < c < 2 and 0 ≤ x < ∞. Then f(0) = 1, lim_{x→∞} f(x) = 0 and

\begin{aligned}f′(x) & = 2^{(b−c)x} + (x + 1)2^{(b−c)x}(b − c)log 2 \\ & = 2^{(b−c)x}(1 + (x + 1)(b − c)log 2). \end{aligned}

Thus

f′(0) = 1 + (b − c)log 2 > 1 − log 2 > 0,

and f has a maximum at x_{0} with

(x_{0} + 1)(b − c) = −\frac{1}{log 2}.

Therefore

max f = (x_{0} + 1)2^{(b−c)x_{0}}

= \frac{1}{(log 2)(c − b)}2^{(b−c)(x_{0}+1)}2^{(c−b)} = \frac{2^{(c−b)}}{c − b} \frac{2^{−1/log 2}}{log 2} .

Since 2^{−1/log 2} = e^{−1} , we have

f(x) ≤ \frac{1}{c − b} \frac{2^{c−b}}{e log 2} < \frac{1.062}{c − b}.

By the above and (3.9) we have

(3.10)  \frac{1}{|g(w) − z|} ≤ ∑_{k=0}^{∞} 2^{b+2}af(k)2^{kc}|w|^{−2^{k}} < \frac{17a}{c − b} ∑_{k=0}^{∞} 2^{kc}|w|^{−2^{k}}.

On the other hand, by Lemma 3.2,

(3.11)  \begin{aligned}∑_{k=0}^{∞} 2^{kc}|w|^{−2^{k}} & ≤ ∑_{k=0}^{∞} ( ∑_{n=2^{k}}^{2^{k+1}−1} n^{c−1}(|w|^{−1/2})^{2^{k+1}}) \\ & ≤ ∑_{k=0}^{∞} ( ∑_{n=2^{k}}^{2^{k+1}−1} n^{c−1}(|w|^{−1/2})^{n}) \\ & = ∑_{n=1}^{∞} n^{c−1}(|w|^{−1/2})^{n} < 4(1 − |w|^{−1/2})^{−c}. \end{aligned}

Therefore by (3.10) and (3.11) we have

(3.12)  \frac{1}{|g(w) − z|} ≤ \frac{68a}{c − b}(1 − |w|^{−1/2})^{−c}.

Since

1 − |w|^{−1/2} = \frac{1 − |w|^{−1}}{1 + |w|^{−1/2}} ≥ \frac{1 − |w|^{−1}}{2} = \frac{1}{2}(\frac{|w| − 1}{|w|}),

by (3.12) we have

|g(w) − z| ≥ m(1 − \frac{1}{|w|})^{c}

for all z ∈ E and 1 < |w| < ∞, where m = (c − b)/272a.

Lemma 3.13. Suppose that g maps B^{∗} conformally onto D^{∗} with g(∞) = ∞. Then

(3.14)  \frac{1}{4} \frac{d(g(w), ∂D^{∗})}{|w| − 1} ≤ |g′(w)| ≤ 4\frac{d(g(w), ∂D^{∗})}{|w| − 1}

for 1 < |w| < ∞.

Proof of Lemma 3.13. For the first half of (3.14), fix w_{0} ∈ B^{∗} and let

G′ = {z ∈ D^{∗} : |z − g(w_{0})| < d(g(w_{0}), ∂D^{∗})}  and  G = g^{−1}(G′).

Then G and G′ are proper subdomains of C. Let

h(ζ) = d(g(w_{0}), ∂D^{∗}) ζ + g(w_{0})

for |ζ| < 1. Then h is a conformal mapping of B onto G′ with h(0) = g(w_{0}). Since g^{−1} ∘ h is an analytic and univalent function of B onto G, by applying the Koebe distortion theorem [P1, p. 22] to g^{−1} ∘ h we have

(3.15)  \frac{1}{4}|(g^{−1} ∘ h)′(0)| ≤ d(g^{−1} ∘ h(0), ∂G).

Since g′(w_{0})(g^{−1} ∘ h)′(0) = d(g(w_{0}), ∂D^{∗}), by (3.15) we have

(3.16)  \frac{1}{4} \frac{d(g(w_{0}), ∂D^{∗})}{|g′(w_{0})|} ≤ d(w_{0}, ∂G).

Thus since d(w_{0}, ∂G) ≤ |w_{0}| − 1, by (3.16) we obtain

(3.17)  \frac{1}{4} \frac{d(g(w_{0}), ∂D^{∗})}{|w_{0}| − 1} ≤ |g′(w_{0})|.

Next for the second half of (3.14), fix w_{0} ∈ B^{∗} and let

G_{1} = {w ∈ B^{∗} : |w − w_{0}| < |w_{0}| − 1}  and  G′_{1} = g(G_{1}).

Then G_{1} and G′_{1} are proper subdomains of C. Let

h_{1}(ζ) = (|w_{0}| − 1)ζ + w_{0}

for |ζ| < 1. Then h_{1} is a conformal mapping of B onto G_{1} with h_{1}(0) = w_{0} . Since g∘h_{1} is an analytic and univalent function of B onto G_{1}′, again by applying the Koebe distortion theorem [P1, p. 22] to g ∘ h_{1} , we have

(3.18)  \frac{1}{4}|(g ∘ h_{1})′(0)| ≤ d(g ∘ h_{1}(0), ∂G′_{1}).

Since (g ∘ h_{1})′(0) = g′(w_{0})(|w_{0}| − 1), by (3.18) we have

(3.19)  \frac{1}{4}|g′(w_{0})|(|w_{0}| − 1) ≤ d(g(w_{0}), ∂G′_{1}).

Thus since d(g(w_{0}), ∂G′_{1}) ≤ d(g(w_{0}), ∂D^{∗}), by (3.19) we obtain

(3.20)  |g′(w_{0})| ≤ 4\frac{d(g(w_{0}), ∂D^{∗})}{|w_{0}| − 1}.

Therefore we obtain (3.14) from (3.17) and (3.20).

Proof of Theorem 1.9. For a fixed w ∈ B^{∗}, we choose z ∈ E such that

|g(w) − z| = d(g(w), ∂D^{∗}).

Then by Lemma 3.13 and Lemma 3.5 we have

|g′(w)| ≥ \frac{1}{4} \frac{d(g(w), ∂D^{∗})}{|w| − 1} = \frac{1}{4} \frac{|g(w) − z|}{|w| − 1}

≥ \frac{m}{4}(1 − |w|^{−1})^{c} \frac{1}{|w| − 1} = \frac{m}{4} \frac{(1 − |w|^{−1})^{c}}{(1 − |w|^{−2})} \frac{1 + |w|^{−1}}{|w|}

= \frac{m}{4}(1 − |w|^{−2})^{c−1} \frac{1 + |w|^{−1}}{(1 + |w|^{−1})^{c}} \frac{1}{|w|}

≥ m m_{1}(1 − |w|^{−2})^{c−1}

for 1 < |w| < \sqrt{2} + 1, where m is a constant in Lemma 3.5 and m_{1} = 10^{−1} .

432 Kiwon Kim

Remark 3.21. With Lemma 1.2, Theorem 1.3, Remark 1.5 and Theorem 1.9 we can summarize the following facts:

Suppose that E is a bounded continuum in C with tr(E) = 1 such that D^{∗} = \overline{C} \ E is connected. Then

(1) (1.4) always holds with b = 2.

(2) (1.4) holds with 1 ≤ b < 2 if E is the closure of a John disk.

(3) (1.4) holds with 1 ≤ b < 2 if

|g′(w)| ≥ m(1 − |w|^{−2})^{b−1}

for 1 < |w| < \sqrt{2} + 1.

(4) (1.4) holds with 1 ≤ b < 2 only if for each constant c, b < c < 2,

|g′(w)| ≥ m(1 − |w|^{−2})^{c−1}

for 1 < |w| < \sqrt{2} + 1.

Therefore, by Remark 3.21 (3) and (4), we have the following characterization of a bounded continuum which satisfies Hypothesis 1.8 in terms of the normalized exterior conformal mapping condition.

Corollary 3.22. (1.4) holds for some 1 ≤ b < 2 if and only if there exists a constant c, 1 ≤ c < 2,

|g′(w)| ≥ m(1 − |w|^{−2})^{c−1}

for 1 < |w| < \sqrt{2} + 1. Here b and c depend on each other.

References

[A] Ahlfors, L.V.: Complex Analysis. - McGraw-Hill, 1979.

[AGH] Anderson, J.M., F.W. Gehring, and A. Hinkkanen: Polynomial approximation in quasidisk. - Differential Geom. Complex Analysis, 1985, 75–86.

[C] Cheney, E.W.: Introduction to Approximation Theory. - McGraw-Hill, New York, 1966.

[GH] Gehring, F.W., and W.K. Hayman: An inequality in the theory of conformal mapping. - J. Math. Pures Appl. 9, 1962, 353–361.

[GHM] Gehring, F.W., K. Hag, and O. Martio: The quasihyperbolic metric in John domains. - Math. Scand. 65, 1989, 75–92.

[GO] Gehring, F.W., and B.G. Osgood: Uniform domains and the quasihyperbolic metric. - J. Analyse Math. 36 , 1979, 50–74.

[J] Jaenisch, S.: Length distortion of curves under conformal mapping. - Michigan Math. J. 15, 1968, 121–128.

[P1] Pommerenke, C.: Univalent Functions. - Vandenhoeck and Ruprecht, Göttingen, 1975.

[P2] Pommerenke, C.: On the derivative of a polynomial. - Michigan Math. J. 6, 1959, 373– 375.

Received 5 September 1994