Some remarks on the value distribution of meromorphic functions

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Some remarks on the value distribution of meromorphic functions

SAKARI TOPPILA

Institut Mittag-Leffler, Djursholm, Sweden and University of Helsinki, Finland 1

1. Introduction

1. Let E be a closed set in the complex plane and f a meromorphic function outside E omitting a set 3". We shall consider the following problem: I f E is thin, under w h a t conditions is F thin, too? I n Chapter 2 we consider the case when E and F are of Hausdorff dimension less t h a n one. In Chapter 3 E and F are countable sets with one limit point, and in Chapter 4 E is a countable set whose points converge to infinity, f is entire, and F is allowed to contain at most one finite value.

2. Sets of dimension less than one

2. Let f be meromorphic and non-constant outside a closed set E in the complex plane. I t is known t h a t if the logarithmic capacity of E is zero then f cannot omit a set of positive capacity, and if E has linear measure zero then f cannot omit a set of positive (1 ~- e)-dimensional measure. I f the dimension of E is greater t h a n one then there exists a non-constant function f which is regular and bounded outside E. Carleson [13 has proved t h a t there exists a set E of positive capacity such t h a t if f omits 4 values outside E t h e n f is rational. We consider the following problem: L e t E be of dimension less t h a n one. Can f omit a set whose dimension is greater t h a n t h e dimension of E?

We denote b y Dim (A) the H a u s d o r f f dimension of a set A, and let dim (A) be the dimension of A obtained b y using coverings consisting of discs with equal radii. For example for usual Cantor sets these dimensions are equal. We have the following answer to our question:

THEORE~ 1. Let E be a closed set with dim (E) < 1. I f f is merornorphic and non-constant outside E and omits F then Dim (F) _< dim (E).

The proof will be given in 3 a n d 4.

1 This research was done at the I n s t i t u t Mittag-Leffler. The author takes pleasure in thanking Professor L e n n a r t Carleson for helpful suggestions.

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ARKIV ~'6R M.A_TE1Vf/kTIK. Vol. 9 No. 1

3. I t does n o t m e a n a n y essential r e s t r i c t i o n t o a s s u m e t h a t m E F , E C {z : Iz[ < 1}, a n d t h a t f is n o n - r a t i o n a l . I n o r d e r t o p r o v e T h e o r e m 1 it is s u f f i c i e n t t o p r o v e t h a t for a n y ~, d i m ( E ) < ~ < 1, a n d a n y R , 0 < R < m, we h a v e D i m ( B ) < a w h e r e B = 2 ' 1 3 { w : ]w I < R } . L e t t h e s e ~ a n d /~ b e chosen. W e d e f i n e

U(a , r) = {z : ]z - - a I < r} .

T h e n we can choose a s e q u e n c e r= w i t h lira r= = 0 a n d coverings

n - - ~ c o

N,

(J U(a~, r ) ~ E

v ~ l

li:m N . r ~ = 0 . (1)

r t - - > o o

such t h a t

F o r a n y a E B we define

f~ is r e g u l a r in {z : 3 < Izl < 4}.

(a E B) for a n y f i x e d

for a n y a E B . W e a s s u m e t h a t siderations. L e t D

f~(z) - - 2~ri (f($) - - a)($ - - z) "

]z] < 4 a n d t h e r e f o r e f~(z) ~ 1/(f(z) - - a). W e set G = B e c a u s e fa(z) a n d 1 / ( f ( z ) - a ) a r e c o n t i n u o u s f u n c t i o n s o f a

r~ < 1/2 for a n y n. L e t n be f i x e d in t h e following con- be t h e c o m p o n e n t o f t h e c o m p l e m e n t of

N=

I.J U(a~, 2 r )

w h i c h contains t h e p o i n t a t infinity. T h e b o u n d a r y o f D consists o f simple closed curves. T h e s e can be d i v i d e d i n t o c o n t i n u o u s c u r v e s y~, v = 1 , 2 . . . . , N , , such t h a t t h e l e n g t h o f a n y y~ is a t m o s t 4zr~. T h e n we get for z C G , a E B ,

fa(Z) 1 1 N~ f dr ] Nn 1

7v

w h e r e % ( a ) = rain I f ( z ) - a L. I t follows f r o m (2) t h a t t h e r e exists a c o n s t a n t

z E y v

fi > 0 n o t d e p e n d i n g on t h e choice o f n such t h a t Nn

v = I

for a n y a E B '

z E G a n d B is c o m p a c t , t h e r e exists fll > 0 such t h a t

f~(z) 1

sup > ~1 (2)

= ~ G f ( z ) - a

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SOME R E ~ A R K S ON T H E V A L U E D I S T R I B U T I O N OF MEROMORPHIC I~UI~CTIONS

W e n e e d t h e following

LI~MMA. T h e r e exists a n absolute constant K > 4 such that i f o , ( a ) <_ ~ (q > O) .for s o m e a E B then o~(b) > K e f o r a n y b E B - - U ( a , 2 K ~ ) .

Proof. T h e l e n g t h o f y~ is a t m o s t 4zr~ a n d U ( $ , r = ) f3 E = 0 for a n y E 7~. T h e l e m m a follows f r o m S c h o t t k y ' s t h e o r e m .

4. L e t 2, ~ < 2 < 1, be chosen. W e choose a p o s i t i v e i n t e g e r k such t h a t

> a(1 + Xk) . (b)

L e t q o = N , . F o r m---- 1 , 2 . . . . , k we set

~,~ ~ 2"r~q,,_lf1-1 , (c)

qm = (P,~/r,) ~ , (d)

a n d let p ~ be t h e i n t e g e r d e f i n e d b y qm --< P , < q , ~- 1. L e t Qk+l be d e f i n e d b y (c) a n d qk+l --= Pk+l ---- 1. I t follows f r o m t h e s e d e f i n i t i o n s t h a t for 1 < m < k -r 1

~m = MmrnN~n m-1 (e)

w h e r e M ~ is a p o s i t i v e c o n s t a n t d e p e n d i n g o n l y on /5. I t does n o t m e a n a n y essential r e s t r i c t i o n t o a s s u m e t h a t -]~n ~ GO as ~--~ OO b e c a u s e o t h e r w i s e E consists o f a f i n i t e n u m b e r o f points. T h e r e f o r e it follows f r o m (e) t h a t we c a n a s s u m e t h a t em+~ < e~ for a n y m.

L e t m, l < m < k A - 1 , be fixed. I f possible, we choose b m , ~ E B such t h a t t h e i n e q u a l i t y %(bz, 1) ~< ~ is satisfied a t least for p ~ d i f f e r e n t v a l u e s o f v. W e set C~, ~ = U(b~, 1 , 2 K e z ) w h e r e K is t h e c o n s t a n t o f t h e lamina. I n t h e s a m e m a n n e r , s t a r t i n g w i t h t h e set

s--1 B - - U C m , p

p=l

(s > 1) we define t h e disc Cm, s. L e t this m e t h o d yieId t h e discs C~ ....

s - - - l , 2 . . . Sin. T h e n it follows f r o m t h e l e m m a t h a t S,~ < Nn/qm.

L e t us suppose t h a t t h e r e exists

k + l S m b E B - - U U C m , p 9

r n = l p = l

T h e n o,(b) > ~k+l for each v a n d ~m+l < %(b) < ~m (1 < m < k) is satisfied at~

m o s t f o r p ~ - 1 d i f f e r e n t v a l u e s o f v. T h e r e f o r e i t follows f r o m (e) t h a t

N= N~ P l - 1 P k - - 1 k+l fi

Z <_ - - + - - + . . . + - - <_ y < S/ro.

v = l e l ~2 ~k-I- 1 m = l

T h i s is a c o n t r a d i c t i o n to (a) a n d so k+l Sm

B e D

U C,~,p.

m = l p = l

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ARKIV le6R MATEMATIK. Vol 9 No. 1

I t follows f r o m (d) a n d (e) t h a t t h e radii o f ^{C m , P} s a t i s f y t h e i n e q u a l i t y

k+s k+s N,Q~ A ~s+;~k~ ~.

5 Sm(2Ke ,,,)~ ~< "AS ~ - - < " ' 2 ' ' n " n

m = S m = S q m - -

w h e r e As a n d .4 2 are positive c o n s t a n t s n o t d e p e n d i n g on t h e choice o f n. F r o m (t) a n d (b) it follows t h a t

1 + 2k~,

N . , . < (N.r~) s+~'k --~ 0

as n - - > ~ . T h e r e f o r e D i m ( B ) < 2 . This is t r u e for a n y 2 > a a n d so :Dim (B) < a. This completes t h e p r o o f o f T h e o r e m 1.

3. Countable sets

5. L e t A a n d B be t w o c o u n t a b l e sets whose p o i n t s c o n v e r g e t o i n f i n i t y . I f B is g i v e n t h e n it is a l w a y s possible t o c o n s t r u c t A such t h a t t h e r e exists a m e r o m o r p h i c f u n c t i o n o m i t t i n g B o u t s i d e A. I n f a c t , we t a k e a n e n t i r e f u n c t i o n f a n d set A = f - l ( B ) . I n this m a n n e r , it is e a s y t o c o n s t r u c t A such t h a t t h e r e exists a c o u n t a b l e f a m i l y o f e n t i r e f u n c t i o n s o m i t t i n g B o u t s i d e A. T h e n t h e following q u e s t i o n arises: Is t h e f a m i l y o f t r a n s c e n d e n t a l e n t i r e f u n c t i o n s o m i t t i n g

o u t s i d e A a l w a y s a t m o s t c o u n t a b l e ? T h e o r e m 2 gives a n e g a t i v e answer.

THEOREM 2. Given a n y countable set B = {bn} w i t h lira b n = ~ then we can construct a countable set A ~ (a,} w i t h lim a~ ~ ~ such that there exists a n o n - countable f a m i l y o f transcendental entire f u n c t i o n s o m i t t i n g B outside A .

Proof. L e t a 0 # 0 a n d a 0 g B. W e shall choose i n d u c t i v e l y a sequence {f.}

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

a o " ~ (fn(Z)) ~ (1)

n ~ S

(e~(1 - - e~) = O) converges u n i f o r m l y in h o u n d e d d o m a i n s for a n y choice of t h e sequence {e,}.

L e t f l ( z ) = z a n d r s = 1 + [bs[. L e t f~ a n d G be d e f i n e d for v--~

I , 2 , . . . , n - - 1 (n > 1). W e d e n o t e b y G, t h e f a m i l y of t h e p o l y n o m i a l s

n--S

g(z) = a o ~-[ (f~(z))~

v=S

w h e r e e ~ ( 1 - e ~ ) = 0 for a n y v. W e choose r . > 2rn_s such t h a t g ( z ) r on

~lz[

---- r . for a n y g q G.. T h e n t h e r e exists 8. > 0 such t h a t Ig(z) - - bl > 5. on

~ z l = r . for all b C B a n d g E G . . W e set

M . = m a x { m a x ]g(z) l } gEGn Izl ^{= r n}

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S O M E R E M A R K S O N T H E V A L U E D I S T R I B U T I O N O F M E R O M O R P t I I C F U N C T I O N S 5

a n d A . = [J {z:g(z) E B a n d [z I < r , } .

g E G n

Because lim b~ = co and G, consists of finitely m a n y polynomials, A , contains a finite number of points. We define

= ] - [ (z - a ) "

a E A n

where t~ is the largest multiplicity of the root a of the equations g(z) ~ b for all g e G. a n d b E B. We set f.(z) = 1 --

e,,h.(z)

where e, ~ 0. Let b e B and g e G . . On ]z] ---r, we have

g(z) f~(z) - g(z)] M.e.lh.(z)]

g(z) -- b l ~-- ^{~ .}

Now we see t h a t we can choose the sequence {Q~} such t h a t

g(z) ~ ]'~ (f,(z)) ~, _ b ^{- -} g(z) [

< 1 (2)

on ]z] = r , for all b E B a n d g C G , , and a n y sequence (e,} (n>__2).

L e t F be the family of entire functions defined b y (1). We define A -= U (z : f(z) E B } .

f E F

L e t

f e F

and b E B. We choose n _> 2. We write

f(z) = g(z) T-T (f~(z)) "" (3)

where g E G,. I t follows from (2) a n d Rouch6's theorem t h a t the functions f a n d g have the same number of b-points in IzL < r,. I t follows from t h e construction of the sequence (f~} t h a t the b-points of g lying in [z[ < r, are b-points of f, a n d not of smaller multiplicity. Therefore in lz[ < r,,

f

can take a value of B only at the points of A , a n d we see t h a t

A c U A ~ .

n ~ 2

I f we choose f such t h a t f E G.+~ we get

(A.+I -- A.) fl {z : ]z I < r,} = 0

a n d we see t h a t co is the only limit point of A. Clearly F contains a non-countable set of transcendental entire functions. Theorem 2 is proved.

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.~d~KIV FOR MATE~IATIK. Vol. 9 No. 1 4. Picard sets for entire f u n c t i o n s

6. Following L e h t o , we call a set E in t h e complex plane a P i c a r d set for e n t i r e functions i f e v e r y n o n - r a t i o n a l entire f u n c t i o n omits a t m o s t one finite value outside E. L e h t o [3] has p r o v e d t h a t a countable set E = {a.} whose points converge t o i n f i n i t y is a P i c a r d set for entire functions if t h e points a , s a t i s f y t h e condition

[a,,/an+l[ = O(n-2) .

M a t s n m o t o [5] has p r o v e d t h e same assertion u n d e r t h e condition exp (K/log la.+x/anl)

lim,~.o~sup

log

lan+l[ < O0

(K a positive constant) w h e n ([an]}~=l,2 .... is s t r i c t l y increasing. Winkler [6] has p r o v e d this assertion in t h e case t h a t E is a finite u n i o n of sets whose points s a t i s f y t h e condition

la=+x/a=l

_> q > 1 a n d

{z : e -laz~ < Lz - aI < lal - ~ } fl E = 0

(e > 0, p > 0) for all sufficiently large la[, a E E. W e shall give a n essentially best possible d e n s i t y condition u n d e r w h i c h a countable set is a P i c a r d set f o r e n t i r e functions.

7. W e shall n e e d t h e following results in our considerations. W e define

If'(z) l

e(f(z)) -- 1 + If(z)[ m '

a n d b y h(r) we denote a n a r b i t r a r y positive f u n c t i o n of t h e positive variable r, w i t h t h e p r o p e r t y h ( r ) = O(r) as r--~ ~ . L e h t o [4] has p r o v e d t h e following THEOREM A. Let f be meromorphic in a neighbourhood

of

the singularity z = oo.

I f for a sequence {z,}, limzn = oo and

n - - ~ o o

lim h(Iz.l ) e(f(zn)) = oo

I I . - > o o

then Picard's theorem holds for f in the union of any infinite subsequence of the discs

C.

= { z : Iz - z.I < ~h(lz.I)}

for each e > O.

Clunie a n d H a y m a n [2] h a v e p r o v e d t h e following THEOI~I~M B. ] f f is an entire non-rational function then

[zle(f(z))

limz~| logIz] -- o o .

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S O M E R E M A R K S ON T H E V A L U E D I S T R I B U T I O N O F M E R O N I O R P H I 0 F U N C T I O N S 7

8. N o w we p r o v e t h e following

TREOREM 3. A countable set E ~ (an) whose points converge to infinity is a Picard set for entire functions i f there exists e > 0 such that

[

z : 0 < ] z - - a ~ ] < log l a n [ / l ' l E =

!~ I

O (1) for all sufficiently large n.

Proof. C o n t r a r y t o o u r assertion, let us suppose t h a t t h e r e exists an e n t i r e non- r a t i o n a l f u n c t i o n f o m i t t i n g t w o f i n i t e v a l u e s outside E . T h e n we can assume t h a t

f

o m i t s t h e values 0 a n d 1 in t h e c o m p l e m e n t o f E .

F r o m T h e o r e m A a n d T h e o r e m B it follows t h a t we can choose a sequence {zn}

such t h a t lim z, ---- ~ a n d P i c a r d ' s t h e o r e m holds for f in t h e u n i o n o f a n y infinite s n b s e q u e n c e o f t h e discs Cn ~- U(zn, r,) w h e r e

~[z.I r , - - 8 log [znl "

T h e n C~ c o n t a i n s at least one zero or 1-point of f for s u f f i c i e n t l y large n, s a y for n > nl. I t follows f r o m (1) t h a t U(zn, 4r,) c o n t a i n s a t m o s t one p o i n t of E if n is large e n o u g h , s a y if n > n 2 > n 1. L e t n > n 2. L e t us suppose t h a t Cn c o n t a i n s a 1-point o f f. B e c a u s e

f

has n o zeros in U ( z , , 2rn), it follows f r o m t h e m a x i m u m principle t h a t t h e r e exists a p o i n t ~ o n ] z - zn[ ~ 2rn such t h a t

If(~)l

~ 1. B e c a u s e f has n e i t h e r zeros n o r 1-points in t h e ring d o m a i n r , < I z - - z ~ ] < 4rn, it follows f r o m S c h o t t k y ' s t h e o r e m t h a t If(z)[ < M on ]z - - z~l ---- 2 r , w h e r e M is a n a b s o l u t e c o n s t a n t . T h e n If(z)[ < M in C,. I f C~

c o n t a i n s a zero o f f we consider t h e f u n c t i o n 1 - - f ( z ) , a n d we see t h a t If(z)] < M Jr 1 in C~ for n > n2. This is a c o n t r a d i c t i o n b e c a u s e f o m i t s a t m o s t one f i n i t e v a l u e in t h e u n i o n o f t h e s e discs. T h e o r e m 3 is p r o v e d .

9. W e n o w p r o v e t h a t t h e c o n d i t i o n (1) is b e s t possible.

THEOREM 4. Corresponding to each real-valued function h(r) satisfying the condition h(r) ---> oo as r ---> ~ , there exists a countable set E -~ {an} whose points converge to infinity, which is not a Picard set for entire functions, and which satisfies the condition

z : 0 < I z - - a , l <

for all sufficiently large n.

laol ]

n E ^:

D

h(ia~]) log Ia,] ]

(2)

Proof. I n o r d e r t o p r o v e o u r assertion, we shall show t h a t t h e set o f t h e zeros a n d 1-points of t h e e n t i r e f u n c t i o n

f(z) = ~ (1

^-^-

z/e'-)'-

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A R K I V FOR M A T E M A T I K . V o ] . 9 N O . 1

(t n being positive integers, tn+ 1 ~ 4tn) satisfies the condition (2) if tn tends to infinity with a sufficient rapidity.

Let n ~ 3 . We define D n = { z : Iz-etnl ~ e x p ( t n - - l t n _ l ) } and

n - - I

g(z) = (1 - Z/etn)'n ]--[ (1 --

It is easy to see t h a t if t~ t e n d s to infinity sufficiently rapidly then

If(z) --

g(z) I <_ 88 Ig(z) l

(3)

in D n. On the b o u n d a r y of D~ we have

Lf(z)l >__

89 > 2. Therefore f has only a finite number of 1-points outside the union of the discs D~, ~ _> 3.

Let $~, ~---- 1 , 2 . . . . ,t,, be the 1-points of g. We set

Then C, c D ~ . On the b o u n d a r y rays of C, we have R e g ( z ) : 0 . Then it is easy to see t h a t

]g(z) l <

211 --

g(z) l

at the b o u n d a r y points of D~ and Q , ~ ---- 1 , 2 . . . t,. Now it follows from (3) and Rouch6's theorem t h a t f has exactly t, 1-points in Dn, each C~ containing one 1-point of f. Then the distance between

n--1

two different 1-points of f in D , is at least r----t~ -~ exp ( t , - ~ t~). Because

n--1 v = l

the term ~ t~ does not depend on t~, we can assume t h a t tn is chosen so large t h a t ~=1

Izl

r ~

h(lzl) log ]zl

for any z E D.. Therefore if E = {an} is the set of the zeros and 1-points of f then E satisfies the condition (2) for all sufficiently large n. Theorem 4 is proved.

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SOME REMARKS OH T H E V A L U E DIST1RIBUTION OF MEROMORPHIC F U N C T I O N S 9

References

1. C~LESON, L., A remark o n Picard's theorem. Bull. Amer. Math. Soc., 67 (1961), 142-- 144.

2. CLVNI]~, J. & HAYMAN, W. K., The spherical derivative of integral a n d meromorphie functions. Comment. Math. Itelv., 40 (1966), 117--148.

3. L]~HTO, O., A generalization of Picard's theorem. A r k . Mat., 3 (1958), 495--500.

4. --~)-- The spherical derivative of functions meromorphic in the neighbourhood of a n isolated singularity. Comment. Math. Helv., 33 (1959), 196--205.

5. MATSVMOTO, K., R e m a r k on Lehto's paper ~)A generalization of Picard's theorems). Proc.

J a p a n Acad., 38 (1962), 636--640.

6. W I N K L E R , J . , ~lber Picardmengen ganzer F u n k t i o n e n . Manuscripta Math., 1 (1969), 191-- 199.

Received M a y 15, 1970 Sakari Toppila,

Maalinauhantie I0 B 36, Rajakyl/i,

Finland