6)= sup f(z) CONFORMAL INVARIANTS AND FUNCTION-THEORETIC NULL-SETS.

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CONFORMAL INVARIANTS AND FUNCTION-THEORETIC NULL-SETS.

BY

LARS AHLFORS and ARNE BEURLING of CA.~IBRIDGE, ~ASS, of UPPSALA.

w i. Introduction.

T h e most useful c o n f o r m a l i n v a r i a n t s are o b t a i n e d by solving conformMly i n v a r i a n t e x t r e m a l problems. T h e i r usefulness derives f r o m t h e f a c t t h a t t h e y m u s t a u t o m a t i c a l l y satisfy a principle of m a j o r i z a t i o n . T h e r e is a rich variety of such problems, a n d if we would aim at c o m p l e t e n e s s this p a p e r would assume f o r b i d d i n g proportions. W e shall t h e r e f o r e l i m i t ourselves to a few p a r t i c u l a r l y simple i n v a r i a n t s and s t u d y t h e i r p r o p e r t i e s a n d i n t e r r e l a t i o n s in considerable detail.

E a c h class of i n v a r i a n t s is c o n n e c t e d w i t h a c a t e g o r y of null-sets, which by this very f a c t e n t e r n a t u r a l l y in f u n c t i o n - t h e o r e t i c considerations. A null-set is the c o m p l e m e n t of a r e g i o n f o r which a c e r t a i n c o n f o r m a l i n v a r i a n t degenerates.

I n e q u a l i t i e s b e t w e e n i n v a r i a n t s lead to inclusion r e l a t i o n s b e t w e e n t h e corre- s p o n d i n g classes of null-sets.

T h r o u g h o u t this p a p e r Y2 will d e n o t e an open region in t h e e x t e n d e d z plane, and Zo will be a d i s t i n g u i s h e d p o i n t in t~. Most results will be f o r m u l a t e d for the case z 0 ~ c~, but t h e t r a n s i t i o n to z o = c~ is always trivial. I n some instances t h e l a t t e r case offers f o r m a l advantages.

W e shall consider classes of f u n c t i o n s

f(z)

which are a n a l y t i c a n d single- valued in some r e g i o n t). F o r a g e n e r a l class ~ t h e r e g i o n t~ is allowed to vary with f , b u t the subclass of f u n c t i o n s in a fixed region t~ will be d e n o t e d by ~(t2). F o r ZoE ~ we i n t r o d u c e the q u a n t i t y

6)= sup If'(zo)l.

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102 Lars Ahlfors and Arne /3eurling.

The abbreviations M~, 21f,~(Zo) or M~($2) will be used when no m i s u n d e r s t a n d i n g can result. I t will be assumed t h a t , ( s 2 ) is n o t empty.

The class ~ is said to be

mo~wtonic

if s s implies ~ ( $ 2 ) < ~($2'). By (I) we have t h e n

(2) (s0, $2) =< (s0, $23.

Suppose now t h a t

z ' = h(z)

defines a one to one c o n f o r m a l m a p p i n g of $2 onto a region $2', and set Zo ~ h(z0). W e shall say t h a t the class ~ is

con formally invariant

if

f(z')E

~ ($2') implies

f(h(z))

E ~ ($2) for all such mappings. F o r a con- f o r m a l l y i n v a r i a n t class we have evidently

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M~ (So, $2) = 21T,~ (So, $2')" I h' (So) I .

This can be w r i t t e n in the more s y m m e t r i c f o r m

I4) M~ (So, $2) I d ~o I = M~ ( g , $2) I d So I,

a n d it is seen t h a t the differential

(5)

M,~ (~,

$2) I ~.Zz I

defines a conformally i n v a r i a n t metric in $2. M~ is itself a relative c o n f o r m a l invariant, and this is the type of i n v a r i a n t we shall be mainly concerned with.

Absolute i n v a r i a n t s can be i n t r o d u c e d either as ~he quotient of two relative invariants or by f o r m i n g the c u r v a t u r e of the metric (5).

I f ~ is both m o n o t o n i c and conformally i n v a r i a n t we can combine (2) and (4) to obtain

(6) M (eo, $2')'ld gl =< $2)" Ide0l

whenever z ' = h(z) maps $2 conformally and one to one onto a subregion of $2'.

W e shall refer to (6) as the

weak monotonic property

of M~.

A ~tronger result is obtained if ~ is

analytically invariant.

By this we mean t h a t f ( z ' ) E ~ (s implies

f ( h

(z)) E ~ ($2) whenever h (z') is single-valued a n d a n a l y t i c in $2 w i t h values in $2', regardless of w h e t h e r

h(z)

is u n i v a l e n t or not. Since a n a l y t i c invariance implies conformal invariance the metric (5) will have the same invariance property as before. An analytically i n v a r i a n t class is

eo ipso

monotonic. H e n c e (6) is valid, b u t the s t r o n g e r a s s u m p t i o n implies t h a t (6) holds n o t only for one to one mappings, but f o r a r b i t r a r y analytie mappings of $2 into $2'.

I n this ease we shall say t h a t M,~ has the

strong monotonic property.

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Conformal Invariants and Function-theoretic Null-sets. 103 A class ~ is said to be compact if the following is true: Given any in- creasing sequence of regions ~2~ and functions f = E ~ ( ~ , ) there exists a subse- quence f~,~ which converges to a limit function f E ~(~), ca = ~ ca~, uniformly on

1

every compact subse~ of s For a compact class there is a function in ~(/2) which makes

If(z0)] a

maximum.

Theorem 1. For a monotonic, co~Tformally invariant and compact elaa.r ~ the followi~g holds."

i) i f ~ tends increasingly to ca, then

(7) lim Ms(zo, ~ ) = Ma(Zo, ca);

ii) ~[S(z, ca) is a contiJ~uous fitnction of z;

iii) log ~ls (z , ca) is subharmouic or ~ - oo.

By (e) lira i.~(~o, ca,,) exists and i~ _--> i~ ( , o , a). On th~ other hand, if f,,

is an extremal function in ~(ca,t), the compactness implies

MS (Zo, ~) ~ lim [A (Zo)[ = lim M~ (z o, ca,) and (7) is proved.

To prove the continuity, let f be extremal in ~(~2) for the point z o. Let z~ be another point in ca such that the circle [ Z - - S o [ < 2 ] z ~ - - z o [ i s contained in ca. W e have

and hence

(s)

Let ~; be the subset of

Ma (,~), ca) ~ If(z{,) I lira ~& (--a, ca) > :<~ (to, ca).

consisting of all points whose distance from the boundary is > [Zo -- Zo I, and let ca" be obtained from ca' by the parallel translation which takes z o to Zo. Then p

~ s (t;, ca) = Ma (~o, ca") = Ms (So, ca')

and as Zo-+ so we obtain by (7)

(9) lim MS (Zo, ca) < M s (zo, ca).

Z o ~ Z o

The inequalities (8) and (9) show that M s(to, ~) is continuous.

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104 Lars Ahlfors and Arne Beurling.

Since log 2/f~(z, ~2) is defined as t h e m a x i m u m in a family of s u b h a r m o n i c f u n c t i o n s log I f ' ( z ) l it m u s t itself be s u b h a r m o n i c .

I n all cases t h a t we shall t r e a t it will be seen t h a t 21~r?(z, s c a n n o t vanish at a single p o i n t unless it vanishes identically. I t seems difficult, however, to f o r m u l a t e a simple general p r o p e r t y f r o m which this would follow.

O u r a t t e n t i o n will be focussed on t h r e e basic classes, too'ether with a subclass of each. T h e first two are the class 93 of b o u n d e d f u n c t i o n s and the class of f u n c t i o n s with a b o u n d e d D i r i e h l e t integral. T h e t h i r d class (~ has a more c o m p l i c a t e d c h a r a c t e r i z a t i o n , b u t it will be shown to be r e l a t e d to the classes 93 a n d ~ in a very s y m m e t r i c m a n n e r .

More precisely, the classes 93(.c2) a n d ~ ( ~ ) consist of all single-valued analytic f n n c t i o n s

f(z)

in ~(2 which satisfy the c o n d i t i o n s I f ( z ) l < ~ and

.if dx __<

.(2

respectively.

T h e class ~ (Y2) is defined only with respect to a p o i n t Zo, and consists of all single-valued anMytie f u n c t i o n s

f(z')

in s with the p r o p e r t y t h a t

(J'(z)--f(Zo)) -~

omits a set of values of area >_--~.

T h e c o r r e s p o n d i n g i n v a r i a n t s are d e n o t e d by

21I,~, 21I~

and Me. As f a r as t h e s e i n v a r i a n t s are c o n c e r n e d we can replace 93, ~) a n d @ by t h e subclasses 930, ~?0 a n d @0 of f u n c t i o n s w h i c h vanish at z 0. This is obvious for the classes

and ~, and f o r a f u n c t i o n

f(z) E ~

we need only observe t h a t .f(~) - f (Zo)

I

--f(zo)f(z )

is in 930 while its d e r i v a t i v e at z o is of absolute value ~ [f'(z)]-

Ill a d d i t i o n we shall consider the subclasses @93, ~ 3 and @~, f o r m e d by all u n i v a l e n t (schlieht) f u n c t i o n s in 93, ~ and ~ . I n o r d e r to be sure t h a t t h e s e classes ~re n o t e m p t y , a n d in o r d e r to make t h e c o r r e s p o n d i n g classes @93o, @~o and @(~o compact, we agree in this c o n n e c t i o n to consider c o n s t a n t f u n c t i o n s as univalent. T h e i n v a r i a n t s

3 I ~ , .3i~

and 21l~e are t h e n well defined.

I t is easy to v e r i f y t h a t all six classes are m o n o t o n i c a n d e o n f o r m a l l y in- v a r i a n t . T h e classes 93 a n d (2 are also analytieMly i n v a r i a n t . H e n c e M~ a n d Me have the s t r o n g m o n o t o n i c p r o p e r t y while t h e o t h e r s have only t h e weak m o n o t o n i c p r o p e r t y . T h e classes 930, ~0, @0 and @930, @'~0, @~0 are compact.

W e are t h u s in a position to apply T h e o r e m I to ~11 o u r i n v a r i a n t s .

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Conformal Invariants and Functiou-theoretic Null-sets. 105

(to)

In this p a p e r we shall prove the i n t e r e s t i n g r e l a t i o n s

-/]I~ : 3lee

Since ~ ~ ~ a n d ~ ~ ~ it will follow t h a t the t h r e e d i s t i n c t i n v a r i a n t s satisfy the i n e q u a l i t y

T h e q u a n t i t y 3Ie~ = 3Ie~ will also be identified with t h e m a x i m u m of an in- v a r i a n t tt (Zo, p) of different n a t u r e , defined by m e a n s of e x t r e m a l lengths.

T h e c o m p l e m e n t of a r e g i o n s will be d e n o t e d by E. Conversely, if a closed set E and a p o i n t z 0 o u t s i d e of E are given, the c o m p l e m e n t of E has a u n i q u e c o m p o n e n t t~ which c o n t a i n s z0. We shall say t h a t E is a

~ull-set

of class ~ if ]l~(Zo, /2) is i d e n t i c a l l y zero. To this definition we observe t h a t f o r all classes c o n s i d e r e d above M~ vanishes identically as soon as it vanishes at a point. This is t r i v i a l f o r t h e classes o~ u n i v a l e n t f u n c t i o n s , f o r t h e n the v a n i s h i n g of ?II~

a t any p o i n t m e a n s t h a t the class ~(s c o n t a i n s only c o n s t a n t s . I n view of (Io) the p r o p e r t y will thus n e e d verification only for t h e class ~ .

T h e i n e q u a l i t y (II) implies the inclusion r e l a t i o n s

and it will be s h o w n by examples t h a t these inclusions are proper. I t follows f r o m (~o) t h a t the t h r e e t y p e s of null-sets h a v e a double c h a r a c t e r i z a t i o n , a n d such i n f o r m a t i o n is of course apt to be valuable.

W e close this i n t r o d u c t i o n on the r e m a r k t h a t a g r e a t e r degree of g e n e r a l i t y c~n be a t t a i n e d by i n t r o d u c i n g classes of m u l t i p l e - v a l u e d f u n c t i o n s . As examples we could consider e i t h e r t h e whole class of f u n c t i o n s

f(z)which

can be con- t i n u e d along all p a t h s in s a n d take only values of m o d u l u s ~ I, or the subclass f o r which } f ( z ) l is single-valued and ~ 1. T h e first choice leads to t h e h y p e r b o l i c m e t r i c with c o n s t a n t n e g a t i v e c u r v a t u r e on ~ p r o v i d e d t h a t E has at least t h r e e points. T h e second choice leads to an i n v a r i a n t which for z o = c~

r e d u c e s to the

capacity

of E . T h e capacity is hence a m a j o r a n t of

ilia,

a n d it follows t h a t all our classes of null-sets contain the sets of c a p a c i t y zeros. T h e p r o p e r t i e s of c a p a c i t a r y null-sets are c o m p a r a t i v e l y well k n o w n , a n d this case will n o t be discussed f u r t h e r .

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T h e r e are also i m p o r t a n t i n t e r m e d i a t e metrics, f o r i n s t a n c e the one w h i c h arises f r o m t h e class of A b e l i a n integrals. I t is m e r e l y f o r t h e sake of con- c e n t r a t i o n t h a t we have decided to leave this und similar cases o u t of con- sideration.

w 2. The Invariants 2tI~ and ~ [ e .

This section is d e v o t e d to t h e p r o o f of t h e r e l a t i o n Ms--~ Me. I t is e v i d e n t t h a t every f u n c t i o n in ~0 belongs to t h e class ~0. T h e r e l a t i o n M~ ~ _Mz is h e n c e trivial, a n d only t h e opposite i n e q u a l i t y n e e d be proved.

Assume t h a t

f(z)E~o(t2)

a n d d e n o t e by A t h e set of values w h i c h ] does I n o t take in ~2. A is a closed set, a n d its area

I(A)

is by h y p o t h e s i s ~ ~. W e f o r m the f u n c t i o n

(I3) ~'(2")-- I ( A ) I I I (W = U Jr- ~V).

w

This f u n c t i o n is clearly a n a l y t i c in f2, a n d its d e r i v a t i v e at z o is

9

J j

₄

=s

I f we can show t h a t ]~'(z)] ~ 1 in Y2 t h e i n e q u a l i t y M~ ~ Me will follow.

I t is sufficient to prove t h a t

A

f o r all c o m p l e x a. An auxiliary c o n g r u e n c e t r a n s f o r m a t i o n is obviously allowed, and h e n c e we m a y t a k e a ~ o and assume t h a t

j w

is real a n d positive.

L e t A § be the p a r t of A s i t u a t e d in t h e r i g h t half-plane.

o r d i n a t e s w = r e ~ we have t h e n

I n p o l a r co-

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Conformal I n v a r i a n t s and F u n c t i o n - t h e o r e t i c N u l l sets. 107

( I 4 ) J J w = . .

A a +

Denote by l(r, O) the linear measure of the set of points wE A + with arg w = O and ] w ] ~ r. S e t t i n g l(c~, 0)----l(O) we have first

z 2

ff f )'"

('5) c o s O d r d O = l(O) cosOdO<--<_ ~ l(O)'dO 9

A + 0

2

On the other hand, l(r, O ) < r, a n d i n t e g r a t i o n with a fixed 0 gives

(,.a,.>= f -Z(~

9 2

Hence

2

(16) I ( A ) >--_

89 f z(O)' dO,

2

and by (I4), (I5) a n d (I6) it follows t h a t

f f d u d v < (zI(A)),/, <_ I ( A ) .

j w

A

This is w h a t we w a n t e d to prove. W e have t h u s shown t h a t M.~ (zo, ~) = Me (Zo, s

I f M s (z0, ~ ) = o, every bou~ded f u n c t i o n in s m u s t satisfy . f (z0)= o.

if f ( z ) is n o t constant, it can be written in the form

a n d t h e n

f ( z ) = f(Zo) + ck (z - - Zo) k + . . . , ck # o

But

f ( z ) - - f(Zo)

(~ - - ZO) k - 1

would be bounded with a non-zero derivative. This is a contradiction, a n d we conclude t h a t the class !~ (~]) contains only constants. I t follows t h a t M~ (z, ~) is identically zero. As already pointed out, this observation is i m p o r t a n t when we consider the identical null-classes N~ a n d Ne. The f o r m e r was first considered by Painlev~ [7], and a set E of class N s will be referred to as a Pain- lev~ null-set.

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108 Lars Ahlfors and Arne Beurling,

T h e f o l l o w i n g t h e o r e m is an i m m e d i a t e consequence of the s t r o n g monotonic p r o p e r t y of M~:

T h e o r e m 2. A non-constant meromorphic function, considered on the comple- ment of a null-set of class N~, takes all complex values with the exception of anolher null-set of the same class.

W e i n t e r r u p t to r e m a r k t h a t the c o r r e s p o n d i n g t h e o r e m is of course valid f o r a n y s t r o n g l y m o n o t o n i c class. A l t h o u g h a direct c o n s e q u e n c e of t h e defini- t i o n s this t h e o r e m is very i m p o r t a n t us a s h a r p a n d g e n e r a l c h a r a c t e r i z a t i o n of o m i t t e d sets. F o r g r e a t e r emphasis we shall give it the following s t r i k i n g f o r m u l a t i o n :

T h e o r e m 2'. Let F be any strongly monotonic class of functions, and let a compact set ~ be measured by m,~(E)= M~ (cx~, D), where D is the complement of E. Then any normalized meromorphic function f ( z ) ~- z + c o + cl + ... in .Q omits

z

the values of a compact set T," with m~ (E') ~ m~ (E).

W e r e t u r n n o w to t h e case of Painlev6 null-sets and n o t e the f u r t h e r char- a c t e r i s t i c p r o p e r t y :

T h e o r e m 3. L Suppose that a r E of class Ne is contained in a regio~

D'. Then every analytic and bounded function f ( z ) in D ' - - - E can be continued to an a~alytic function in D'. Conversely, ~f the continuatio,~ is always possible, the set E is of class N,~.

By a s t a n d a r d a p p l i c a t i o n of Cauchy's i n t e g r a l f o r m u l a we can write f ( z ) ----A (z) + A ( z ) , where A ( z ) i s a n a l y t i c in .Q' and A ( z ) is a n a l y t i c in ~, the c o m p l e m e n t of ~:. B u t t h e n f2(z) is bounded, and if E is a null-set it m u s t r e d u c e to a c o n s t a n t , so t h a t f(z) m u s t be a n a l y t i c in D'. T h e converse is obvious,

C o r o l l a r y . The value of lhe invariant M,~ (z 0, D) does not change ?f a null-set of class N~ is removed from 52.

I n fact, the f a m i l y of c o m p e t i n g f u n c t i o n s r e m a i n s t h e same.

1 T h e first precise s t a t e m e n t of t h i s t h e o r e m is difficult to locate, b u t it is i m p l i c i t in t h e w o r k of PAINLEV~; [71"

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Conformal Invariants and Ftmction-theoretic Null-sets. 109 w 3. T h e I n v a r i a n t s 2hr~ and M e e .

W e shall now prove t h a t M s - - Mee. W e wish to p o i n t o u t t h a t this r e s u l t and the m e t h o d

by

which it is proved are previously known, a l t h o u g h n o t exactly in t h e p r e s e n t connection. T h e idea of t h e p r o o f goes back to G r u n s k y [4], and a t h e o r e m by Schiffer [~o] is essentially e q u i v a l e n t with ours. Nevertheless, it is essential f o r o u r purposes to give a n e w version of t h e proof.

T h e classes ~o and ~ o b o t h satisfy the conditions of T h e o r e m I. F o r this r e a s o n it is' sufficient to p r o v e t h e r e l a t i o n 3 I ~ M e e f o r r e g i o n s which can be used to a p p r o x i m a t e an a r b i t r a r y r e g i o n f r o m within. W e are t h e r e f o r e allowed to assume t h a t the r e g i o n .q u n d e r c o n s i d e r a t i o n is b o u n d e d by a finite n u m b e r of a n a l y t i c curves. T h e complete b o u n d a r y , t a k e n in the positive sense with respect to t h e region, will be d e n o t e d by /'.

T h e existence of a u n i v a l e n t f u n c t i o n

I

p ( ~ ) = - - - - + . ( ~ - s0) + .

Z - - Z o

which maps t? onto a r e g i o n b o u n d e d by h o r i z o n t a l slits is well k n o w n . Simi- l a r l y , t h e r e exists a f u n c t i o n

q(z) - - I § b ( z - - Zo) + . . .

- - S o

which maps Y2 o n t o a region b o u n d e d by v e r t i c a l slits.

L e t

f(z)

be any r e g u l a r a n a l y t i c f u n c t i o n in the closed region /2. By a f a m i l i a r f o r m u l a

O ( f , p - - q ) = : . f'(z)(p'(z)--q (z))dxdy-~ ;.

B u t

d23=dp

a n d d ~ = - - d q on F. H e n c e

f f ( d ~ -- dO)= f f((~p + dq)= - - a ~ i f ' (Zo)

1' l '

by t h e residue t h e o r e m , a n d we obtain

(I8)

D(,f, p - q)-= 2 ~:f' (Zo).

F o r

f = p - - q

this f o r m u l a gives

D (p - - q) == 2 7~ (a - - b) a n d we find, i n c i d e n t a l l y , t h a t a - b is r e a l a n d positive.

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The Schwarz inequality

[ D ( f , p - - q) J~ ~ D ( f ) D ( p - - q) now yields

4 ~ I f ' (eo)I ~ ~ 2 ~ (a - - b) D (f), a n d hence D ( f ) ~ z implies

with equality for

(IV)

I t is thus proved t h a t

if' (Zo)] ~ ] / / a --2 b

f ( z ) = P - - q

V2 (6 - b)

1 / ^a b M~ (zo, ~2) = [ /

2

I n fact, any s t a n d a r d a p p r o x i m a t i o n technique can be used to show t h a t (I8) remains valid when f ( z ) is k n o w n to be analytic only in t h e open r e g i o n ~.

W e t u r n now to the class ~ . F o r f u n c t i o n s g ( z ) w h i c h are analytic in the closed region ~2 except for a simple pole a t Zo we i n t r o d u c e the i n t e g r a l

i

1"

I f the pole is missing, I ( g ) is equal to the Diriehlet integral D ( g ) , and in the presence of a pole it can be used as a substitute for D ( g ) . I f g is univalent, - - I ( g ) is t h e a r e a . e n c l o s e d by the image of F, a n d if I_ is of class ~ ( [ 2 ) we

g have hence I(g) ~ -- ~r.

The corresponding bilinear i n t e g r a l can again be evaluated by the residue theorem 9 W e find

(2i) I(g,p + q ) = i f 9 (d~ + d 4) = i f

g ( d p - - d q) = - - ~ c (a - - b),

I ' p

where c is t h e residue of g at z o. In particular,

(22) l ( p + q) = - - 2 z (a - - b).

F r o m the f a c t t h a t

I a - ~ ( p + q) ----D ,q-- + q) _>-o

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Conformal Invariants and Function-theoretic Null-sets.

we obtain by (2I) and (2z)

_ ~ I~(~ ~), S(q)_>- ~l~ -

and if

I ( g ) ~ - 7r

the inequality

A < V a - b

follows, with equality for the function

P + q g = V2 (a-- b)"

The relation

(23)

111

1 / ^a b

2

will hence be proved if we can show that the function p + q is univalent. We shall then have found identical representations (2o) and (23) of M~ and M~e.

In order to investigate the nature of the function p + q we observe that

d q dp

is purely imaginary on F with two simple zeros and two simple poles on each contour. Then ~

dq

cannot vanish at any interior point, for a level curve

~

dq

~ o would have to pass through a pole and there are no such curves be- sides the contours. Since

~dqqa.p : I at z o

we conclude that ~ > o throughout the region. This implies that ~ p

dq

decreases along each contour, and hence

arg ( d p + d q ) = are tg ( ~ d~)

is also decreasing with the total variation - - 2 ~. W e conclude that each contour is mapped on a convex curve, and a standard argument shows that p + q is univalent. Our proof of the relation

M~ (Zo, ~) --- m e e (Zo, ~2) is now complete.

Since

Mee(Zo,

~): is evidently ~

Me(zo, ~2)

we have also proved, in con- junction with (I7) , the inequality

i ~ (~, ~) =< M~ (~, ~).

(12)

112 Lars Ahlfors and Arne l~eurling.

I n t h e i n t r o d u c t i o n we h a v e a l r e a d y r e m a r k e d t h a t ~Ie~ ~ o only if ~ ( ~ c o n t a i n s only t h e c o n s t a n t zero, a n d this p r o p e r t y is of course i n d e p e n d e n t of z o. W e can n o w conclude t h a t t h e class ~ e n j o y s t h e s a m e p r o p e r t y , a n d we can i n t r o d u c e t h e i d e n t i c a l null-classes N z a n d N~e. T h e f o r m e r h a s p r e v i o u s l y been c o n s i d e r e d by N e v a n l i n n a [6] a n d Sario I9]. T h e i d e n t i t y of 2v~ a n d Nz~

c a n be e x p r e s s e d m o r e explicitly as follows:

Theor-em 4? A set E is a ~ull-set of class N~ i f and on 0 i f every region which is confbrmatty equivalent with the complement of E has a complement of zero area.

T h e f o l l o w i n g t h e o r e m is a n a l o g o u s to T h e o r e m 3, a n d it is p r o v e d in the s a m e m a n n e r .

T h e o r e m 5. 3 Every analytic function f ( z ) with 1) (.f) < c~ in ~2" -- E can be extended to an analytic fu~wtion in ~' i f a~d only i f E is a ~ull-set of class ~ .

C o r o l l a r y . The value of M~ (Zo, s does not change i f a ~ull-set of class 2 ~ is removed from .Q.

I t is easy to s h o w t h a t t h e r e l a t i o n s (2o) a n d ( 2 3 ) r e m a i n valid f o r a r b i t r a r y r e g i o n s ~ if p a n d q a r e defined as l i m i t s of t h e c o r r e s p o n d i n g f u n c t i o n s f o r an a p p r o x i m a t i n g sequence of r e g i o n s w i t h a n a l y t i c b o u n d a r y . T h i s r e m a r k leads to t h e f o l l o w i n g c h a r a c t e r i z a t i o n of null-sets of class N u :

T h e o r e m 6. 3 A set E is a null-set of class N~ *f and only i f every univalent function i~2 the complement of E is linear.

I f E is a null-set e v e r y u n i v a l e n t f u n c t i o n c a n be e x t e n d e d to a mero- m o r p h i c f u n c t i o n in t h e whole plane. W e m a y in f a c t a s s u m e t h a t t h e f u n c t i o n h a s a pole o u t s i d e of E , a n d t h e n its D i r i c h l e t i n t e g r a l o v e r a n e i g h b o u r h o o d of E is finite. T h e r e s u l t i n g f u n c t i o n h a s a single pole a n d is h e n c e linear.

Conversely, if E is n o t a null-set, p a n d q c a n n o t b o t h be linear, f o r t h e n t h e y would be i d e n t i c a l a n d we would h a v e a - b----o.

T h e c o n s i d e r a t i o n s of t h i s section are s u i t a b l y s u p p l e m e n t e d by a discussion of t h e q u a n t i t y

The necessity was recently pointed out by MYRBERG [5]. There is no record of the sufficient condition.

Stated and proved in SARIO [91.

s Stated in NEVANLINNA [6] and proved in SARIO [9]"

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" (: 'o: r ----

and in p a r t i c u l a r

Conformal Invariants and Function-theoretic Null-sets. 113 M~(.q, z,.,, -(7-)= sup If(.-',)--f(z~)l

.f E ~ (-'-')

defined with respect to two points zl, z., in L). W e prove first:

T h e o r e m 7. The va~Ushing of M s ( z t , ze, ~) is eqMralent with the identical va~dshing of ~ (zo, -(2-).

I n t h e first place, if M~(zo, -Q)= o t h e class ~ (.(2.) contains only c o n s t a n t f u n c t i o n s a n d Mz(z~, ze, ~) vanishes trivially. T h e converse can be proved us follows: L e t f(2`) be u n i v a l e n t in .(2- and choose any ZoE .Q. T h e f u n c t i o n

f ' (2`0)

f (2`) - - f(2`o) 2` - - 2`0

has a finite Dirichlet integral, and if -Mz (2`~, ze, -(2-) = 0 we m u s t c o n s e q u e n t l y have

,X" (2'0) I

f'

^(2`0) ^I

,/'(z,) - - f(Zo) z, --2`0 f(2`e) --J'(zo) z2 - - Zo

W i t h 2`o as variable this is a d i f f e r e n t i a l e q u a t i o n with l i n e a r solutions. H e n c e all u n i v a l e n t f u n c t i o n s are linear, and by T h e o r e m 6 this implies ]ll~(z0, .0.)=0.

T h e i n v a r i a n t 3.S~(2`1, Ze, t2) can be d e t e r m i n e d explicitly by a m e t h o d com- pletely analogous to the cne used f o r d e r i v i n g the r e l a t i o n (20). W e assume again t h a t the b o u n d a r y F of .(2. is composed by a finite n u m b e r of a n a l y t i c curves. I t is possible to map o by f u n c t i o n s P(2`)and Q ( z ) o n t o regions b o u n d e d by c o n c e n t r i c and r a d i a l slits respectively so t h a t z I is m a p p e d into o and z.~

into oz. We may normalize t h e m a p p i n g s so t h a t b o t h f u n c t i o n s h a v e t h e residue i at 2`e, and we set P'(zl)----A, Q ' ( 2 ` I ) = B .

T h e f u n c t i o n l o g ~ P is a n a l y t i c a n d single-valued in .(2-. F o r any r e g u l a r f u n c t i o n f(2`) in ~ we o b t a i n

log -~ . . . . 7 f d log P ? -- 2

,~(J'c<) -f(~,,)),

1' 1'

F r o m this we derive

39 log = 2 :r log ~ .

lY(2`D - - f ( < ) I " < =" 2-7~ i log A .~- D ( f ) , a n d h e n c e D ( f ) ~ r~ implies

9 - 642136 A c t a m a t h e m a t i c c t . 83

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114 Lars Ahlfors and Arne Beurling

[/'(z_~) - f ( z , ) [ < ~ log

with equality for a multiple of log P Q. I t follows that

(24) )I~ (zl, z~, ~) = log --.

B

The result remains true for an arbitrary region ~2 provided we define P and Q as limits of the corresponding slit-functions for a sequence of approxi- mating regions. We conclude that M s = o if and only if the functions are identical.

It could also be proved that

V-ffQ

is univalent and maps t2 on a region whose exterior has maximum logarithmic area.

In w 6 we shall give an interesting interpretation of the relation (24)in the case where 1!: lies on the circle ] Z ] = I.

w 4- The I n v a r i a n t s d~le~ and ,~/e~.

The equality of dlle~ and Me.~ will result from comparison with a third invariant, defined by means of extrcmal lengths. An account of the theory of extremal lengths is under preparation, but since it cannot yet be referred to we shall list below the definition and main properties of this notion.

Let {7} denote a family of rectifiable curves in a region ~2. Consider the class of non-negative functions Q(z) in t2 for which the quantities

L,~, {7} = i n f f o l d z l

7 7

are defined and not simultaneously o or co. The least upper bound

with respect to this class is called t h e

extremal length

of the family {7}. The value of ;~ {7} does not depend on t h e region Y2, but very frequently the family {7} will be defined with reference to a specific f2.

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Conformal Invariants and Function-theoretic Null sets. 115 I t is easy t o see t h a t 1/7} is a c o n f o r m a l invarian~ in t h e sense t h a t any c o n f o r m a l one to one m a p p i n g of .q will t r a n s f o r m {y} into a f a m i l y {7'} with

z it'}

= ~

{r}.

The following properties are immediate consequences of the definition:

L e m m a 1.

then

I f two families {7} and {7'} are such that every 7 contains a 7', z It) >= z {r'}.

L e m m a 2. / f the families {7~} aJ~d {7~1 cover disjoint pointsets, and i r a third family {7} is such that a,ery Y contains a Y~ and a Y~, then

z It} --> z {r,) + z {r~}.

L e m m a 3. I f the families {71} and {7.~} cover disjoi~*t pointsets, and i f every 71 and 7.~ is contained hz a curve 7 of a third f a m i l y {7}, then

I I I

z l r ) = z l r , } zlr~}

L e m m a 4. The extremal length of the f a m i l y of curves which join the sides of length a in a rectangle with the sides a, b is - . _ab

L e m m a 5. The extremal lenoth of the f a m i l y 03" curves which separate two

~ircles I~1 = ," a , d I~1 = R > r is equal to ~ ~ / l o g ~ .

r

I n t h e p r e s e n t c o n n e c t i o n we shall only consider e x t r e m a l l e n g t h s w h i c h are defined in a very special way. L e t Y2 be a region, z o a p o i n t of ~2 a n d /~o a subset of t h e c o m p l e m e n t E of ~2. W e d e n o t e by {7}r t h e class of simple closed curves in ~2 which separate z o f r o m Eo while m a i n t a i n i n g a d i s t a n c e ~ r f r o m zo. T h e e x t r e m a l l e n g t h ~.{7}r will t e n d to zero w i t h r. B u t if r ' < r i t follows f r o m L e m m a s 3 and 5 t h a t

o r

W e c o n c l u d e t h a t

I I I ]'

- - > - - + - - l o g ~ z / 7 } ~ , = z b'}~ z ~ r

2:71:

Z{rl,

- - + l o g r ~ Z {r}~' + log r . 2 ~

2.'r

(Zo, Eo) lira I

r ~ 0 r

(16)

exists. T h e differential e l e m e n t

, (.~0, t~o) ] d z0 I

is c o n f o r m a l l y i n v a r i a n t f o r a p r o p e r definition of the t r a n s f o r m of L o.

I t follows f r o m L e m m a I t h a t tt(zo, Eo) is a n o n - d e c r e a s i n g f u n c t i o n of the set E 0 a n d a non-decreasing f u n c t i o n of s W e shall call #(z0, t o ) the perimeter of E 0 with respect to t h e region ~2 and t h e c e n t e r Zo. F o r a circle J z - - z o l < R all subsets of t h e c o m p l e m e n t have the p e r i m e t e r I / R .

I t is obvious t h a t t h e value of #(Zo, ~0) depends only on t h e set of com- p o n e n t s of E which c o n t a i n points of _E o, and n o t on the i n d i v i d u a l points w i t h i n a c o m p o n e n t . T h u s t h e p e r i m e t e r of a single p o i n t is equal to t h e p e r i m e t e r of t h e c o m p o n e n t to w h i c h it belongs. T h e p e r i m e t e r of a p o i n t p is d e n o t e d by

~t(Zo, p). F o r a simply c o n n e c t e d r e g i o n ~t(.Zo,p) has only one value, a n d f o r so = oo t h i s value equals t h e capacity of E . I n t h e g e n e r a l case, ~t(cr E ) =

= cap E .

W e shall prove:

T h e o r e m 8. The invaria~zts ~/le~ and ~[~3 are both equal to the maxin~um of

~t (,~o, 2) ) for p E E .

W e suppose first t h a t s is b o u n d e d by a finite n u m b e r of a n a l y t i c c o n t o u r s I ' ~ , . . . , F,,~. T h e n ~t(Zo,p) has only n values, one f o r each c o m p o n e n t of t h e c o m p l e m e n t . T h e r e exists a f u n c t i o n .fk(z) which maps ~2 on a r e g i o n b o u n d e d by t h e u n i t circle, c o r r e s p o n d i n g to Fk, and n - I c o n c e n t r i c circular slits; we suppose t h a t t h e c e n t e r c o r r e s p o n d s to Zo. F o r a r e g i o n of this sort it is easily p r o v e d t h a t t h e p e r i m e t e r of t h e o u t e r c o n t o u r is exactly i, r e g a r d l e s s of the n u m b e r and location of the slits. By c o n f o r m a l i n v a r i a n c e we have h e n c e

~, (z o , r~.) -- I.f,; (Zo)] =< M| (Zo, s and we have p r o v e d thai~

m a x !t

(Zo, p) =< M ~ (Zo, s

Conversely, suppose t h a t f(z) maps s on a subregion of [ w [ ~ - I and t h a t f(Zo)-~ o. T h e image of s has a definite o u t e r c o n t o u r w h i c h c o r r e s p o n d s to a Fk, a n d by a p p l i c a t i o n of L e m m a I a n d e o n f o r m a l i n v a r i a n c e we obtain at once

If' =<, r,0.

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Conformal Invariants and Function-theoretic Null-sets. 1 17 H e n c e

M ~ (Zo,/2) =< m a ~ . (Zo, p), a n d we h a v e p r o v e d t h a t max tt(z5, p)=-71le~(Zo, s

L e t us now consider a m a p p i n g of /2 by a f u n c t i o n of class ~ . T h e i m a g e r e g i o n has a g a i n a definite o u t e r c o n t o u r w h i c h we suppose c o r r e s p o n d s to / ' >

W e replace /2 by its image s u n d e r w = f k ( z ) ; /2k is a u n i t circle with con- c e n t r i c slits.

F o r ,fe | (/2k), set

L(,)= f If'l Idwl

luq=r

w h e n e v e r [w[ = r does n o t c o n t a i n any slit, a n d

D ( , , ) = f f If'l ~ ~Z,,d~ (w = u + i~)

f o r all r. By t h e S c h w a r z i n e q u a l i t y we have first L (,') ~ <= 2 ~ ," D ' (,')

f o r all n o n - e x c e p t i o n a l r. On the o t h e r h a n d , since t h e image of [ w l < r will always have the i m a g e of ]w] = r as its o u t e r c o n t o u r , t h e i s o p e r i m e t r i c inequality yields

L (r)" ~> 4 z D (r).

H e n c e

D'(r) > 2,

1 ) (,-) = r

and i n t e g r a t i o n f r o m r o to I gives

1) (to) =< 1) (~) ,-~ ~ ~,.~.

L e t t i n g ro t e n d to o we conclude t h a t I t ' ( o ) ] < i. I n t e r m s of t h e o r i g i n a l region ~q it is t h e n p r o v e d t h a t

a n d since all the f u n c t i o n s fk (z) are of class ~5~ we find M| = m a x lfi (Zo){ = M e ~ .

I n t h e g e n e r a l ease we a p p r o x i m a t e /2 with an i n c r e a s i n g sequence of regions /2,~ with a n a l y t i c b o u n d a r i e s . W e write /~(z0, p ) w h e n t h e i n v a r i a n t is t a k e n with r e s p e c t to Q,~. W e h a v e trivially

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118

and hence

Lars Ahlfors and Arne Beurling.

t* (Zo, P) < ,",, (~0,1')

sup tt (go, p) ~ lim max ~t,~ (Zo, p) = Me,~ (Zo, 32).

The opposite inequality can be proved directly. Suppose t h a t w = f ( z ) with /(Zo)=O maps 32 on a subregion of I l< I . W e can a sequence of points wn = f ( z ~ ) which tends towards the infinite component of the complement of s Let p be a Limit point of the sequence z,. Then any curve 7 which separates z o from p has an image which separates o from Iwl = I, and we conclude immediately t h a t

**If' (Zo) J < t* (Zo, p)**

and consequently the equation

~3/e~ (Zo, 32) = max ~ (Zo, p)

holds for arbitrary 32. The relation Me~ = M ~ for arbitrary s follows of course directly by a limit process.

w 5. Further Characterization of the Null-sets N ~ .

I f E 1 and E~ are disjoint compact sets in or on the boundary of a region 32, the extremal distance ~ ( E 1 , E2) between the sets with respect to the region 32 is by definition the extremal length E {7} of the family of curves 7 which join E 1 and E~ within 32.

We stated in Lemma 4 of w 4 t h a t the extremal distance between opposite sides of a rectangle R is equal to the ratio ~ of the sides. Suppose now t h a t a a compact set E is removed from R. Then, by Lemma I of w 4, the extremal distance between the sides with respect to R - - E is known to be > a/b. W e claim t h a t the sign of equality will hold for all rectangles R if and only if E is a null-set of class N~.

We may assume t h a t the rectangle R lies symmetrically with respect to the coordinate axis, the sides of length a being parallel to the x-axis. The ratio a

is the extremal length between the vertical sides. As in w 3 we shall ap- proximate the complement 32 of E by regions 32n with analytic boundary, and introduce the functions .p,~(z) for z o = vo. I f E is a null-set of class N~ we know t h a t lim pn (z) = z.

n ~ 0 0

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Conformal Invariants and Function-theoretic Null-sets. 1 1 9 F o r large n p . (z) will m a p t h e p e r i m e t e r of R o n a q u a d r i l a t e r a l w h i c h differs very little f r o m R. W e m a y h e n c e find a,, and

bn,

t e n d i n g to a and b, such t h a t i ~ p = ( z ) l < a= on the v e r t i c a l sides of J7 a n d = -

13P,,(z) l

> b,, = - - o n t h e h o r i z o n t a l

2 2

sides. L e t t h e r e c t a n g l e with sides a , a n d b, be d e n o t e d by R,,. E v e r y c u r v e which joins t h e v e r t i c a l sides of Rn c o n t a i n s t h e image of a curve j o i n i n g the v e r t i c a l sides of R w i t h i n R - - E . By L e m m a I, w 4, we can h e n c e conclude t h a t t h e e x t r e m a l d i s t a n c e ZR-E w i t h r e s p e c t to R - - E satisfies

a ~ l

and p a s s i n g to the l i m i t we o b t a i n

ZR-Z <= a/b.

This proves t h a t t h e e x t r e m a l distance does n o t c h a n g e w h e n a set E of class N~ is removed.

T o prove t h e converse, assume n o t only t h a t ZR-E----b' b u t also t h a t t h e e x t r e m a l distance ~R-E b e t w e e n t h e h o r i z o n t a l sides of R has t h e value b L e t

a

~ : s ( z ) be an a r b i t r a r y u n i v a l e n t m a p p i n g of ~2 w i t h a pole a t r I t will t r a n s f o r m t h e p e r i m e t e r of R into a simple closed c u r v e whose i n t e r i o r can be m a p p e d in t u r n by a f u n c t i o n w ~ ~ (z) o n t o a r e c t a n g l e /7' of d i m e n s i o n s

a', b'.

C o n f o r m a l i n v a r i a n c e a n d L e m m a I, w 4, lead to opposite inequalities f r o m which we c o n c l u d e t h a t

a t g

- - .

b-7--~

Choose 0 = ~ - in R - - E . F o r e v e r y curve 7 whieh joins t h e v e r t i c a l sides of R w i t h i n R - - E we shall t h e n have

By t h e definition of Z~-~: t h e area

f e l d z l >

a'.

7

r e c t a n g l e R m u s t h e n c e be m a p p e d o n t o a n

# 2 # r

a

/2,R-e=

a b.

This m e a n s t h a t t h e m a p of R - - E will fill o u t all of R ' e x c e p t f o r a set of m e a s u r e zero, a n d since t h e d e r i v a t i v e I~b'(z) I is h o u n d e d away f r o m zero on t h e image of R - E , it follows t h a t

s(z)

m u s t m a p ~ o n t o a r e g i o n whose comple-

(20)

merit is of zero m e a s u r e . T h i s is t r u e f o r an a r b i t r a r y u n i v a l e n t m ~ p p i n g , h e n c e f o r t h e m a p p i n g p + q of w 3, a n d h e n c e E is of class ~ .

T h e p r o o f could be modified so as to a p p l y to a r b i t r a r y q u a d r i l a t e r a l s , a n d in f a c t to a r b i t r a r y e x t r e m a l distances. W e shall t h e r e f o r e a n n o u n c e o u r r e s u l t in t h e f o l l o w i n g f o r m :

T h e o r e m 9 ) A set E is a ~ull-set of class N~ i f a~d o~ly :if the removal of E does no.t change extremal dista~ces.

W e h a v e a s s u m e d , so f a r , t h a t E is c o n t a i n e d in t h e o p e n r e c t a n g l e /~.

T h e r e s u l t r e m a i n s of c o u r s e t r u e w h e n t h e i n t e r s e c t i o n of R w i t h an a r b i t r a r y E of class N~ is r e m o v e d , a l t h o u g h t h e p r o o f is n o t so t r i v i a l as it m i g h t seem.

L e t R ' be a c o n c e n t r i c r e c t a n g l e w i t h sides a' ~ a a n d b' < b. Since E is totally d i s c o n n e c t e d , it is possible to find a c u r v i l i n e a r q u a d r i l a t e r a l R " which is con- t a i n e d in t h e r e c t a u g l e w i t h sides a', b a u d c o n t a i n s t h e r e c t a n g l e w i t h sides a, b', a n d whose p e r i m e t e r does n o t m e e t /~. I t encloses a c o m p a c t s u b s e t E "

of E , a n d we h a v e h e n c e )~R"-E" ~ hE". On t h e o t h e r h a n d , by t w o a p p l i c a t i o n s of L e m m a I, w 4, we o b t a i n

~R-Z <: ).W'-Z" = ~F:' ~ ~R' = a'/b'

be chosen a r b i t r a r i l y n e a r to a/b we find ~.Z-E <= a/b as a n d since a'/b' can

desired.

W e r e m a r k also t h a t the o t h e r h a l f of T h e o r e m 8 has been p r o v e d in s l i g h t l y s t r o n g e r f o r m , f o r we h a v e s h o w n t h a t E is of class N~ as soon as t w o p a r t i c u l a r e x t r e m a l d i s t a n c e s are u n c h a n g e d . I t c a n be p r o v e d in a t r i v i a l m a n n e r t h a t ZR-E ~ a/b if t h e p r o j e c t i o n of E on t h e v e r t i c a l sides is of m e n sure zero. T h i s a c c o u n t s f o r t h e sufficient c o n d i t i o n in

T h e o r e m 10. A set E is of class 5 ~ i f its project]oils in two orthogonal di- rections are of linear measure zero. On the other hand, ~f E is of class N~ any two poi~#s in the complement s can be joined by a curve in s whose length differs arbitrarily little from the distance between the points.

T h e n e c e s s a r y c o n d i t i o n is easily p r o v e d . I f t w o p o i n t s h a v e a d i s t a n c e in .q w h i c h is s u p e r i o r to t h e i r d i s t a n c e in the plane, it is c l e a r t h a t a t h i n rec- t a n g l e R can be c o n s t r u c t e d s u c h t h a t t h e d i s t a n c e of t w o sides is g r e a t e r in R - - E t h a n in R. T h i s i m p l i e s ).Jt-E ~ ).~, a n d h e n c e /~ c a n n o t be of class N ~ .

1 A related theorem in different terminology and connection is found in GI~0TZSCIa [3]-

(21)

Conformal Invariants and Function-theoretic Null.sets. 121 w 6. Linear Sets.

I n this section we shall always choose z 0~- oo. W e can t h e n t h i n k of the i n v a r i a n t s M,~, dlI~ a n d Me~ as f u n c t i o n s of a c o m p a c t set E which does n o t divide t h e plane.

T h e r e is a classical r e l a t i o n b e t w e e n M~ and t h e l i n e a r m e a s u r e of the set E . More precisely, we shall d e n o t e by / / t h e g r e a t e s t l o w e r b o u n d of the t o t a l l e n g t h of a system of closed curves 7 w h i c h s e p a r a t e E f r o m cx~ a n d we shall p r o v e t h a t

(2S) 21I <= ! . . 4 .

2 7 t

This is an i m m e d i a t e c o n s e q u e n c e of C a u c h y ' s t h e o r e m . g u l a r a n d of absolute value G I in ,Q we have i n d e e d

I f f ( z ) - - c_ + . . . is re-

z

7 7

a n d t h e r e l a t i o n (25) follows at once.

W e shall consider s e p a r a t e l y the case w h e r e E lies on a s t r a i g h t line, f o r instance on t h e real axis. I f t h e l i n e a r m e a s u r e of E is L we have ~/--~ 2 L and (25) implies

(26) M~ =< ~ L .

A n i n e q u a l i t y in t h e opposite d i r e c t i o n is o b t a i n e d by c o n s i d e r i n g t h e f u n c t i o n

dx L +

E

I t is i m m e d i a t e l y seen t h a t I~ t f ( z ) l < zr, a n d t h e f u n c t i o n

e 2 - - I

.f(z) e 2 -~ i

is h e n c e of class ~ with the first coefficient -.L W e have t h u s 4

(z7)

^-L ^.

4

(22)

I n particular, we may conclude:

T h e o r e m 11.1 A linear set is of class ~ i f and oMy i f it is of li~lear mea-

8$~Ye z e Y o .

More generally, we may consider a set E on an analytic curve 7. We can still prove:

T h e o r e m 11'. A set E on an analytic curve is of class ]~% i f a~d o~dy ~if it is of leugth zero.

This is proved by showing t h a t every bounded function which is analytic in a region ~ ' - - E , where ~ ' is an open neighborhood of E, is analytic in ~ ' if and only if /~ is of linear measure zero. B u t it is clearly sufficient to prove the corresponding local statement, which follows from the fact that every point on 7 has a neighborhood which can be mapped conformally so t h a t 7 w i l l cor- respond to a segment of the real axis. I n order to apply T h e o r e m 3 it is nec- essary to choose the neighborhood so that its boundary does not intersect E.

I f E is totally disconnected this is always possible, and if E contains an arc n e i t h e r 21I~ nor the linear measure can be zero.

Let us now find a bound for M~ when E is a compact set on th'e real axis and has given length L. This problem is not quite easy and needs some preparations. Consider first a function f(z) of the form

oO

where ~(t) is of summable square and vanishes outside /s By an application of the Fourier integral and the P a r s e v a l relation, we find this relation for D ( f )

o ~ o o

//'

I ) ( f ) - ~ Jr q~(s) -- 9(t)l~ds dt -- H(9)

-

- - o 0 - - 0 0

which holds w h e t h e r both sides are finite or infinite. Conversely, if D ( f ) i s finite, f(z) is generated by a f u n c t i o n ~ with H ( q ~ ) = D ( f ) .

Since H(~o) ~ H(I~I), we may conclude t h a t the extremal function f of the class ~ ( t ] ) is generated by a ~ which is real and ~ o on E. L e t now ~0" be

i T h i s t h e o r e m is d u e t o DEN JOY [2].

(23)

Conformal Invariants and Function-theoretic Null-sets. 123 the even, symmetrically decreasing and equi-measurable function to ~0, and se~

kn(s) = Min ( z s -~, n). Thus H ( r is the limit as ,~t-~ co of the expression

c r ~ o O o O

Y/

((~v~(s) +

qv~(t))k.(s - t ) d s d t - - 2 // ~ ( s ) ~ ( t ) k . ( s - - t ) d s d t = A,,(~)--B.(q;).

- - ~ - - c o ~ o 0 - - o o

Obviously .4~ (~) = A~ (f*), while

B , (~o) <_ B,,

(~0") according to a rearrangement theorem due to Hardy, Littlewood and Polya (see e. g. Inequalities, Cambridge I934, Theorem 38o). Thus H(~o) ~ H(~0*) and

D ( f ) ~ D ( f )*

follows, where f * is the function generated by ~0". Since ~0" vanishes outside the segment E*

limited by the points _+

L/2,

the function f * must be holomorphic outside /~*

and we conclude that M e (E)--< Me ( E * ) = - - . L 4

Theorem 12.

For a linear set of leugth L we have the string of i~equalities

(28) _Mz,~ _--< Me _--< L __< M~ _--__ L

4 z

I t is interesting to note that M~ is smallest while Me,~ and M e are largest when E consists of a single segment. In the next section we shall show that there is no lower bound for Meu or M e in terms of L.

Theorem 13. /~br

linear sets M ~ a~2d Me are simultaneously positive or---o.

According to Theorem 8, the perimeter tt vanishes for every boundary point of ~ if Me~----o. This property is obviously invariant under schlicht mappings of t) onto 2 ' and thus implies that the complement of Y2' is always totally disconnected. I f in particular E is linear, both slit functions p and q must degenerate. Thus a ---- b -~ o and M~ ~ o follows.

W e shall now give a more precise characterization of linear sets of class Ne. This is mos~ easily

done

for sets E which lie on the unit circle ]z]---- I.

We begin by supposing that E consists of a finite number, of closed arcs a;. The complement of E on the circle is denoted by E ' and consists of open arcs fli. According to formula (24) of w 3 the invariant M e (o, c~, Y2) is deter mined by the functions

P(z)

and

Q(z)

introduced in that section. Obviously

P(z) ~ z ,

making A----I, while

Q(z)

must satisfy the relation

B

(24)

f r o m which it follows t h a t [ Q ( z ) [ = B ! on t h e arcs fli. On t h e o t h e r h a n d , we k n o w also t h a t ~ l o g ] Q ( z ) [ ~ - o 0 on hi, a n d f o r this r e a s o n B - I Q ( z ) can be c o n t i n u e d f r o m ]z[ > I across t h e a,. to a f u n c t i o n QI(z), defined a n d single- valued out.side of t h e arcs fii, w h i c h satisfies

-(;)

^{= Q,}

W e conclude t h a t t h e f u n c t i o n

v(z) = ( - log Q, (z) l - . l o . B )

is r e g u l a r and h a r m o n i c outside of /~' e x c e p t f o r a l o g a r i t h m i c pole at c~ which is such t h a t V(z) + log I z l v a n i s h e s f o r z :- c ~ V(z) is t h e n t h e e q u i l i b r i u m p o t e n t i a l of E ' w i t h t h e c o n s t a n t v a l u e - - 88 log B on t h e set. T h e c a p a c i t y of E ' is hence B t a n d it follows f r o m (24) t h a t

V '

(29) .21/3 (o, o o ~) = 2 log cap E'"

T h i s r e s u l t can i m m e d i a t e l y be c a r r i e d o v e r to t h e ease of an a r b i t r a r y closed set E on t h e circle. T h e c o m p l e m e n t E ' h a s t h e n an inner c a p a c i t y , defined as t h e least u p p e r b o u n d of t h e c a p a c i t i e s of closed s u b s e t s of E ' . I t follows by a t r i v i a l l i m i t i n g process t h a t (29) r e m a i n s valid, p r o v i d e d t h a t cap E ' is inter- p r e t e d as t h e i n n e r c a p a c i t y .

F r o m (29) we derive t h e following~ criterion:

T h e o r e m 1 4 } A closed set E on the unit circle is of class N~ i f and o~ly i f the inner capacity of'its complement is equal to I.

I t will be noted, of course, t h n t t h i s does n o t i m p l y t h a t t h e set E is of zero c a p a c i t y .

T h e r e is a m o r e g e n e r a l t h e o r e m whose p r o o f we shall omit.

T h e o r e m 14'. A closed set on an analytic arc is of class N~ i f a~d only i f the inner capacity of its complement is equal to the capacity of the arc.

w 7- Special Sets.

I n o r d e r to show t h a t t h e classes Ne~, N~ a n d 5 ~ are all d i s t i n c t we m u s t e x h i b i t a set which is in Ne~ b u t n o t in N~ a n d a s e t in N~ which is n o t in Nu.

1 Certain results in de POSSEL [8] a r e related to this theorem.

(25)

Conformal I n v a r i a n t s and Function-theoretic Null-sets. 125 W e shall also show t h a t t h e r e are l i n e a r sets of p o s i t i v e m e a s u r e in ~Y~ a n d N ~ . ] n m o s t of t h e cases t h e e x a m p l e s will be g e n e r a l i z e d C a n t o r sets.

o9 O0

L e t

{qi}l

be a sequence of r e a l n u m b e r s 0 <

qi

< I a n d {ni}~ u sequence of positive i n t e g e r s . W e shall c o n s t r u c t a c o r r e s p o n d i n g l i n e a r C a n t o r set E({q~}, {n~}) as a closed subset of t h e u n i t i n t e r v a l E 0 : 0 ~ t ~ t. T h e first step is to divide E 0 in 2 n 1 + I s u b i n t e r v a l s , t h e odd ones of l e n g t h a 1 =

q~/(n I +

I) a n d t h e e v e n ones of l e n g t h b 1 = ( I -

ql)/n~;

f o r s i m p l i c i t y t h e y will be r e f e r r e d to as a~-intervals a n d b~-intervals, a n d t h e u n i o n of t h e closed a~-intervals is d e n o t e d by E 1. I n t h e n e x t step each a l - i n t e r v a l will be s u b d i v i d e d in 2 ~ + I a l t e r n a t i n g a~- a n d b,~-intervals of l e n g t h a2 a n d b~ r e s p e c t i v e l y . T h e s e l e n g t h s are chosen so t h a t t h e a~-intervals cover a p r o p o r t i o n q~ of t h e al-intervals, a n d t h e u n i o n of all a~-intervals is d e n o t e d by E~. T h e p r o c e s s is r e p e a t e d a n d we o b t a i n a n e s t e d sequence of sets _E~ ~ E 2 ~ . . . whose p r o d u c t

E = EI E 2 . . .

is t h e C a n t o r set

E({qi},

{hi}) w h i c h we set o u t to define. T h e l e n g t h of E is [ [ q , . I t is p o s i t i v e if a n d only if ~ ( I - - q i ) < o o .

1 1

W e shall first d e r i v e a sufficient c o n d i t i o n f o r E to be a null-set of class

~Vz~. By L e m m a 3 a n d 5 of w 4 this will be t h e case if each p o i n t of E can be s u r r o u n d e d by a sequence of d i s j o i n t a n n u l i c, w h i c h do n o t m e e t E a n d whose d e c r e a s i n g r a d i i r : a n d r: s a t i s f y t h e c o n d i t i o n

(30) log r

1 r v

L e t us fix o u r a t t e n t i o n on a p o i n t t E E , I t b e l o n g s f o r e a c h k to a cer- t a i n ak-interval w h i c h we shall d e n o t e b y

ak(t).

W e s u r r o u n d t by a n n u l i c e n t e r e d a t t h e m i d p o i n t of

ak(t)

w h i c h p a s s t h r o u g h t h e bk-intervals c o n t a i n e d in ak-1 (t); s o m e of t h e s e m a y i n t e r s e c t t h e r e a l axis in only one b~:-interval. I n o r d e r to m a k e sure t h a t t h e a u n u l i do n o t m e e t E a n d are all d i s j o i n t we a g r e e to include only t h o s e a n n u l i whose i n n e r r a d i u s is a t l e a s t e q u a l to ak + bk w h i l e t h e o u t e r r a d i u s is a t m o s t e q u a l to bk-1. I t is clear t h a t such a n a n n u l u s c a n n o t i n t e r s e c t a n y ak-x~interval o t h e r t h a n

ak-l(t)

a n d h e n c e c a n n o t m e e t E . More- over, a n a n n u l u s of t h e k + 1:st g e n e r a t i o n c a n n o t m e e t a n a n n u l u s of t h e k:th g e n e r a t i o n , f o r a c o m m o n p o i n t would a t once be a t a d i s t a n c e ~ ak + bk f r o m t h e c e n t e r of

ak(t)

a n d a t a d i s t a n c e ~ bk f r o m t h e c e n t e r of

ak+l(t)

w h i c h is i m p o s s i b l e since t h e t w o c e n t e r s h a v e a m u t u a l d i s t a n c e < ak.

(26)

1 2 6 Lars Ahlfors and Arne Beurling.

T h e s m a l l e s t a n n u l u s of t h e k:th g e n e r a t i o n w h i c h c a n satisfy t h e i m p o s e d c o n d i t i o n s h a s r a d i i ~ak + b~. a n d ~ a~ + 2 bk, a n d t h e s e r a d i i a r e i n c r e a s e d by ak + bk a t e a c h step. T h e n u m b e r vk of t h e l a s t p e r m i s s i b l e a n n u l u s is t h e r e f o r e d e t e r m i n e d by t h e c o n d i t i o n

a~ + b~ + yr. (ak + b~.) < b~.-~.

2

B u t ak + bk < ak-~/nk a n d sufficient to t a k e

b k - i :> I - - - - qk_~ a~-~ > (I - - q~._~)ak_~. I t is t h e r e f o r e qk-~

= - q k - , ) ] -

w h e n e v e r this n u m b e r is positive. W e note, m o r e o v e r , t h a t

i)

= - - l o g i - - - - >

t~k - - + b~, + v ( a k + bk

2

v -t- I ak + b~

> -I ~ - .q k

v §

H e n c e t h e a u n u l i of t h e k:th g e n e r a t i o n c o n t r i b u t e to t h e sum (3 o) an a m o u n t g r e a t e r t h a n

z 3

w h e r e t h e f a c t o r in f r o n t is > log vk + 2 T h e whole c o n t r i b u t i o n is t h u s g r e a t e r 4

t h a n

(I - - qk) log

4 a n d we conclude t h a t ]g is of class Nz~, w h e n e v e r

n k ( I q~--1)

(3 I) Z (i - - qk )log

1 4

diverges. T h i s c o n d i t i o n does n o t c o n t r a d i c t t h e c o n v e r g e n c e of ~ (I - - qk), a n d

1

we h a v e p r o v e d :

(27)

Conformal Invariants and Function-theoretic Null-sets. 127 Theorem 15. There exists a linear set of positive measure which is a null-set of class Ne~.

The Cartesian product of two identical linear Cantor sets of positive length is a 2-dimensional Cantor set of positive area. As such it will certainly not be of class _ ~ . However, this set is again of class Nz~ whenever the series (3 I) diverges. The proof is the same as above, except t h a t it is more convenient to replace the circular rings by quadratic frames. In (3 o) we let r:/r', be the ratio of the outer and inner dimensions of the frames and it is elementary to show t h a t the divergence of the series (3 o) is still a sufficient condition for the set to be of class Nz~.

Theorem 16. There exists a set of class 2Ve~ of positive area, and hence not of class . ~ .

I t remains only to construct a linear set of positive measure which is of class 5r~. Such a set cannot be of class 2 ~ and therefore also serves to show t h a t Nu is a proper subclass of N~.

To this purpose we shall make use of Theorem I4, and our object is thus to construct a closed set /~ on the unit circle which is of positive length while the inner capacity of its complement E ' is equal to I.

L e t us first observe t h a t the inner capacity of a finite number of open arcs is equal to the capacity of the closed arcs. I t is also wellknown t h a t the capacity of an arc of length 4/2 and radius I is sin I/2. L e t now u(z) be the equilibrium potential of the arc

E;: Io-ool< l (rood

Then I-u(z ?~) is found to be the equilibrium potential of the set E~: [ n b ) - - O o I . < 2 / ] t (rood 2 r from which we conclude t h a t

? l

c a p /~'7~ - - V s T n i / ~ = I - - -

while the length of E,'~ is 4/2.

This example proves the existence of open sets with arbitrarily small length and with an inner capacity arbitrarily close to x. Taking a sequence {/~}o of P

such sets with length L , = 4/~n such t h a t