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