# On subadditivity of analytic capacity for two continua

Nobuyuki Suita

^{1}01 Jan 1984-Kodai Mathematical Journal (Department of Mathematics, Tokyo Institute of Technology)-Vol. 7, Iss: 1, pp 73-75

About: This article is published in Kodai Mathematical Journal.The article was published on 1984-01-01 and is currently open access. It has received 10 citation(s) till now. The article focuses on the topic(s): Thesaurus (information retrieval) & Subadditivity.

Topics: Thesaurus (information retrieval) (58%), Subadditivity (56%), Analytic capacity (50%)

## Summary (1 min read)

### ll/llsi

- The following problem was first raised by Vituskin [6] .
- This shows that the constant M possibly depends on positions of two sets.

### 2. Extremal problem.

- For simplicity's sake the authors write K lf K 2 for its boundary components which have the same analytic capacities as the original continua.
- Since the analytic capacity is invariant under a conformal mapping with the normalization (4), the authors have Since the analytic capacity is equal to the capacity for a continuum [7] , they obtain the assertion from the minimum property in Theorem 1.

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N.

SUITA

KODAI

MATH.

J.

7 (1894), 73-75

ON

SUBADDITIVITY

OF

ANALYTIC

CAPACITY

FOR TWO CONTINUA

Dedicated

to

Professor Mitsuru Ozawa

on his

60th birthday

BY

NOBUYUKI

SUITA

1.

Introduction.

Let

K be a

compact

set in

the complex plane

C

and

let

Ω(K)

be

the connected component

of

C—K containing the point

at

infinity.

The

analytic capacity γ(K)

of K is

defined

by

r(/O=sup

|/'(oo)|.

ll/llsi

Here

f(z) is a

holomorphic function

in

Ω(K) whose expansion

at the

point

at

infinity

is

given

by

f(z)=f'{°o)/z+a

s

/z*+~'

and

11/11 denotes the supnorm

of /.

The

following problem

was

first

raised

by

Vituskin [6]. For two disjoint

compact

sets

K

λ

and

K

2

, is

there an absolute constant

M

such that the following

semiadditivity holds

We

can

find this problem

and

partial answers

in

many articles

and

books

[1,

2, 3, 6,

7].

It

was shown

by

MeΓnikov

[3]

that

if K

1

and

K

2

are

separated

by

an

analytic curve

c,

then there

exists

a

constant M(c) depending

on c for

which (1) holds. This shows that the constant

M

possibly depends

on

positions

of two sets.

In

the present paper

we

shall establish the following subadditive inequality

(2)

riK^Kj^γiKj+ΠKz)

for arbitrary disjoint continua

K

λ

and

K

2

. To

prove (2)

we

shall

solve

an

auxi-

liary extremal problem.

2.

Extremal

problem.

Let Ω be a

doubly connected domain containing the

point

at

infinity.

We

suppose that its boundary components are continua

K

Y

and

Received March 22, 1983

73

74 NOBUYUKI SUITA

K

2

. Let EF be the family of univalent functions φ defined on Ω with a normal-

ization

(3) φ(z)=z+ajz+••• at co.

We denote by φ(Kj) the boundary component of the image domain φ{Ω) under φ

which corresponds to Kj (j=l, 2). Our problem is to minimize Cap ^(ϋΓJ+Cap φ(K

2

)

within £F, Cap φ(Kj) being logarithmic capacity of φ{K

3

). We prove

THEOREMI.

The minimum of the quantity Cap 0(AΊ)+Cap φ(K

2

) is

attained

by the con

formal

mapping

φ

0

which

maps

Ω

onto

a

parallel

slit

domain Δ

whose

boundary

lies

on a

straight

line.

Proof.

Since capacity is invariant under rigid motions, we consider the same

problem within a slightly larger class § of univalent functions φ whose expansion

admits

(4)

φ{z)

= e

%θ

z+a<>+ajz+- at co.

There

exists

a unique radial

slit

disc mapping φ(z) in 5 such that φ{K

±

) is a

disc

I

w

I

Sr, say and ψ(K

2

) is a line segment a^w^b (a>r). Then Cap φ{K^)=r

and

we know that r is equal to the minimum of Cap

φ{K^).

For these facts see

Sario and Oikawa ([5], pp. 148-158). By an elementary mapping S

r

{w) —

r(w/r+r/w), ψ(K

λ

) is mapped onto a line segment [—2r, 2r] with the same

capacity r. Then ψ{K

2

) is mapped onto a new segment [α',

b'~\.

We show that

Cap[α',

6']=:(&'—αO/4 is the minimum of Cap 0(iQ within £F. In fact by a

parallel displacement, [α', fr'] is moved onto

[—2r',

2r'], r

f

—{b

f

—a

f

)/A. The

conformal mapping S^

1

maps the last domain onto another radial

slit

disc, K

2

corresponding to the disc

\w\^r

f

.

Hence r' is the minimum of Cap

φ(K

2

).

Thus

r+r

/

is the desired minimum within §.

S

r

°ψ

is clearly an extremal function.

It

is easy to see that by composition of elementary mappings an extremal func-

tion

φ

0

can be constructed in £F as stated in the theorem, which completes the

proof.

Remark. The uniqueness of the extremal function in 9" is deduced from

that

of a radial

slit

disc mapping. Indeed the radial

slit

disc mapping ψ with

lϊm

z

^φ(z)/z—l is a unique extremal mapping minimizing radii within the family

of univalent functions φ

satisfying

Iim

2

^oo^(^)/z=l and that φ{K

λ

) is a disc

centered

at the origin. It is not difficult to derive the uniqueness directly by

means

of the method of extremal metrics.

3.

Subadditivity.

We prove

THEOREM

2. // K

x

and K

2

are mutually

disjoint

continua, then the

inequality

(2)

holds.

ON

SUBADDITIVITY

OF

ANALYTIC CAPACITY

75

Proof.

If Ω(Kj)aΩ(K

k

) (jΦk), the inequality is trivial. Otherwise Ω(K^JK

2

)

forms a doubly connected domain. For simplicity's

sake

we write K

lf

K

2

for its

boundary components which have the same analytic capacities as the original

continua.

Let φ

0

be the extremal mapping for this domain in Theorem 1.

Pommerenke

showed that for a compact set on a straight line its analytic capacity

is equal to a quater of its total length [4]. Since the analytic capacity is

invariant under a conformal mapping with the normalization (4), we have

Since the analytic capacity is equal to the capacity for a continuum [7], we

obtain

the assertion from the minimum property in Theorem 1.

REFERENCES

[ 1 ]

BRAXNNAN,

D.

A.

AND

J. C.

CLUNIE,

Aspect

of

contemporary complex analysis,

Academic Press (1980).

[2]

GARNETT,

J.,

Analytic capacity

and

measure, Springer Lecture Notes

in

Math.

297 Springer-Verlag (1972).

[ 3 ]

MEL'NIKOV,

M.

S.,

Estimate

of the

Cauchy integral along

an

analytic curve, Math.

Sbornik

71

(1966), 503-515 (Russian), Amer. Math.

Soc.

Translation

ser. 2, 80

(1969), 243-256.

[4]

POMMERENKE,

C,

Uber

die

analytische Kapatitat, Arch,

der

Math.

11

(1960),

270-277.

[ 5 ]

SARIO,

L.

AND

K.

OIKAWA,

Capacity functions, Springer-Verlag (1969).

[6]

VITUSKIN,

A.

G.,

The

analytic capacity

of

sets

in

problems

of

approximation

theory, Uspehi

Mat.

Nauk

22

(1967), 141-199 (Russian), Russian Math,

surveys

22 (1967), 139-200.

[7]

ZALCMAN,

L.,

Analytic capacity

and

rational approximation, Springer Lecture

Notes

in

Math.

50,

Springer-Verlag (1968).

DEPARTMENT

OF

MATHEMATICS

TOKYO

INSTITUTE

OF

TECHNOLOGY

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01 Jan 1968-

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108 citations