On thc Valiron deficiencies of integral functions of infinite order
W. K. H A Y ~
I. Introduction
Let f(z) be meromorphie in the plane. We d e f i n e in the normal w a y the order and the characteristic T ( r , f ) of f(z) and also the quantities re(r, a) and N(r, a) for a n y a in the closed plane. 1)
The Valiron deficiency is defined to be
m(r, a) N(r, a)
A(a) = ,-~lim T ( r , f ) -- 1 -- ,-,~lira T ( r , f ) "
We are concerned in this paper with the question of how large the set of a can be for which A (a, f) > 0 [3, problem 1.2]. For functions of finite order this problem has recently been completely solved b y I~yllengren. He proved [4, Theorem 1]
the following
THEOREM A. Let E be any plane point set. T h e n the following two conditions are equivalent
a) There exists a positive number k and an infinite sequence al, a2, . . . of complex numbers, so that each a E E satisfies the inequality
[a -- anI < exp{-- exp(nk)}
f o r infinitely m a n y n.
b) There exists a real number x, 0 < x < 1 and a meromorphic function f(z) o f finite order i n [zl < ~ , so that
A ( a , f ) > x f o r every a i n E.
I n fact f ( z ) can be chosen to be an integral f u n c t i o n . 1) for the notation see e.g. [5, p. 158].
164 w . K . ~IAYMA:N
F o r functions of infinite order the s i t u a t i o n is r a t h e r different. The strongest result in t h e positive direction is due to Ahlfors [1, see also 5, p. 264], who proved T~EO~]~M B. Suppose that
f(z)
is meromorphic in the plane. T h e n given e > 0, w e have for all a outside a set E of capacity zerore(r, a) < T ( r , f ) 1/~+~ (1.1)
for all sufficiently large r. I n particular A (a) = 0 outside a set of capacity zero.
As H y l l e n g r e n points out, while all sets satisfying his condition a) h a v e c a p a c i t y zero, t h e converse is false, a n d in f a c t sets satisfying a) are m e t r i c a l l y s u b s t a n t i a l l y smaller t h a n general sets of c a p a c i t y zero.
.
I n this paper we shall give examples to prove t h a t for functions of i n f i n i t e order T h e o r e m B is more or less best possible, b y p r o v i n g
T~EOREM 1. Let E be an arbitrary F . set of capacity zero. T h e n there exists a n integral function f(z), such that A(a, f ) ~ 1 for a E E.
This result is a n i m m e d i a t e consequence of t h e following more precise T~EOREM 2. Let q51(r ) and ~ ( r ) be continuous increasing functions of r for r ~ r o, which tend to -}- ~ with r. Let E., be an expanding sequence of compact sets o f capacity zero, having the origin as an isolated point. T h e n there exists an integral function f(z) with f ( O ) = 0, and a sequence r.~--~ ~ with m, such that for
m -~ 1 , 2 , . . . , we have
n(r.~, a , f ) ~ q~l(r.,), a e E . , , (2.1) 2r a , f ) ~_ q~l(r.,) log r,., a C E . , , (2.2) a n d
T(r,~,f) ~_ q52(rm) . (2.3)
W e n o t e t h a t r can t e n d to i n f i n i t y as slowly a n d qS~(r) as r a p i d l y as we please.
T a k i n g for instance Cbl(r ) = log r, ~b~(r) = r, we see t h a t all values a in E = [9 Era, satisfy A (a, f ) ~ 1. I t is also interesting to n o t e t h a t t h e lower g r o w t h of 2V(r, a) for a E E m a y be as slow as we please, subject t o being more r a p i d t h a n ]og r.
Using t h e second f u n d a m e n t a l theorem, we deduce f r o m (2.2) t h a t T(r.,/2, f ) < 4qil(r~) log (r~), m > too,
O N T H E Y A . L I R O N D E F I C I E I ~ C I E S O F I N T E G R A L F U N C T I O N S O F I N F I N I T E O R D E R 165
w h i c h c o n t r a s t s w i t h (2.3). W e also d e d u c e f r o m (2.2) a n d (2.3) t h a t t h e sequence r,, is e x c e p t i o n a l for Nevanlinna?s second f u n d a m e n t a l t h e o r e m . F o r t h a t t h e o r e m implies as r--+ ~ outside a n e x c e p t i o n a l set o f f i n i t e m e a s u r e [5, p. 241]
(q -- 2)T(r,f) ~_ (1 + o(1)) ~ q = l •(r, a~,f)
f o r a n y q d i s t i n c t v a l u e s a,. I f this were t r u e for r~ we s h o u l d d e d u c e q52(r~ ) < (q + o(1))(q - - 2)-lq~l(rm) w h i c h is false in general. T h u s t h e e x c e p t i o n a l set in N e v a n l i n n a ' s second f u n d a m e n t a l t h e o r e m can r e a l l y o c c u r [see 3, p r o b l e m 1.22].
T h e a s s u m p t i o n t h a t Em is c o m p a c t a n d has t h e origin as a n isolated p o i n t is n o real restriction. F o r i f t h e E n are a r b i t r a r y closed sets we m a y w r i t e E'~ for t h e p a r t o f U : ~ I E , in m -1 _~ [z 1 ~ m, t o g e t h e r w i t h t h e origin. I f t h e E , all h a v e c a p a c i t y zero, so does E : , w h i c h is c o m p a c t a n d
E = U==I E : = UZ=, E= O { 0 } . (2.4)
T h u s a n y F , set E c o n t a i n i n g t h e origin can be w r i t t e n in t h e f o r m (2.4). I f we n o w choose for i n s t a n c e q~l(r) --~ r, q52(r ) = # in T h e o r e m 2, we d e d u c e t h a t for e v e r y a in E
N(rm, a) O(r,~ log r , )
- - 2 ----N 0 , a s r , n ~ c O ,
T(rm, f ) r=
so t h a t A ( a , f ) = 1. T h u s T h e o r e m 1 follows i m m e d i a t e l y f r o m T h e o r e m 2.
T h e o r e m 2 also shows t h a t if E is a n y F~ set o f c a p a c i t y zero a n d r q~(r) are t h e f u n c t i o n s of T h e o r e m 2, t h e n t h e r e exists a s e q u e n c e r~ - + co a n d a n i n t e g r a l f u n c t i o n f(z) such t h a t (2.3) holds a n d for a n y a E E we h a v e
N(rm, a) ~ q~l(rm) log r~, m ~_ too(a) 9 (2.5) I n this f o r m t h e r e s u l t is best possible. I n f a c t t h e set o f a s a t i s f y i n g (2.5) for a g i v e n m o is a n i n t e r s e c t i o n o f closed sets a n d so is closed. T h u s t h e set E o f alt a satisfying (2.5) for a g i v e n sequence r~ is a n F~ set. I t follows f r o m a r e s u l t o f N e v a n l i n n a [5, f o r m u l a 18, p. 171] t h a t a n y closed subset o f E , a n d so E itself, m u s t h a v e c a p a c i t y zero if (2.5) holds, as soon as
~5:(r~) - - q~l(r~) log r~ --> + oo, as m --> c o . (2.6) T h u s it is r e m a r k a b l e t h a t once t h e v e r y w e a k c o n d i t i o n (2.6) is satisfied, we do n o t r e s t r i c t t h e set E f u r t h e r b y decreasing q~l(r) or increasing r
3. Some preliminary results
W e c o m p l e t e t h e p a p e r b y p r o v i n g T h e o r e m 2. I n o r d e r t o do this we n e e d to r e p r o d u c e a s i t u a t i o n in a f i n i t e disk for a sequence o f values r --~ rm in t h e plane.
W e n e e d t w o s u b s i d i a r y results.
1 6 6 w . K . H A Y ~ A N
L E M ~ 1. Let E be a compact set of capacity zero, p a positive integer and x a positive number. Then there exists c h, such that x ~ c~ ~ 10x and a function
F(z) = alz + %+1 zv+l + . . . , (3.1) regular in Iz[ ~ 1, univalent in [z] ~ % / 2 - 1, having unbounded characteristic and assuming no value of E more than once in ]z] < 1.
W e assume i n i t i a l l y t h a t E does n o t m e e t t h e real axis, e x c e p t p e r h a p s a t w = 0 . L e t E o be t h e set E w i t h t h e p o i n t s 0, T x , oo, a d d e d w h e r e x is a positive n u m b e r . L e t _R be t h e i n f i n i t e covering surface o v e r t h e c o m p l e m e n t o f E 0. W e cut a c o p y o f _R f r o m x t o ~- co along t h e real axis t h u s o b t a i n i n g t w o surfaces R i, /t~ a n d a n o t h e r c o p y o f / ? f r o m - - x to - - co o b t a i n i n g t w o surfaces _R 3 a n d /?4, all of which are s i m p l y c o n n e c t e d . L e t R 5 be t h e p l a n e cut f r o m x t o co a n d f r o m - - x t o - - co a l o n g t h e real axis a n d let _R 0 be o b t a i n e d b y joining R1, R 2 to R s on t h e segments (x, oo), a n d _R 3 a n d _Ra t o R~ along t h e segments ( - - o o , - x), so t h a t R 0 is a R i e m a n n surface c o n t a i n i n g n o n e o f t h e p o i n t s
=F x, co in a n y sheet a n d c o n t a i n i n g p o i n t s o v e r E e x a c t l y once, n a m e l y in t h e s h e e t 17 5. T h u s _R 0 is s i m p l y c o n n e c t e d , a n d since /?0 does n o t c o n t a i n t h e p o i n t s
~= x, oo, / t o is h y p e r b o l i c . T h u s we m a y m a p Tz] < 1 (1, 1) e o n f o r m a l l y o n t o R 0 b y a f u n c t i o n
Fo(z) = biz -~ b~z ~ Jr . . . w h e r e b i ~ O.
T h e f u n c t i o n 2'0(z ) n e v e r assumes t h e values :F x, oo, a n d so is s u b o r d i n a t e t o t h e f u n c t i o n G(z) w h i c h m a p s IzZ ~ 1 o n t o t h e i n f i n i t e covering surface S o v e r t h e p l a n e w i t h t h e s e 3 p o i n t s r e m o v e d a n d satisfying
G ( O ) = O , G ' ( 0 ) > 0 .
This l a t t e r f u n c t i o n m a p s z = 1, i, - - 1, - - i o n t o w = x, i oo, - - x, - - i co so t h a t t h e s h e e t _R s c o r r e s p o n d s t o a ~>quadrilatera]~> Q in t h e u n i t disk b o u n d e d b y 4 q u a r t e r circles joining these p o i n t s (1, i), (i, - - 1), ( - - 1, - - i) a n d ( - - i, 1) a n d o r t h o g o n a l to Iz I = 1. Clearly Q contains t h e disk Iz[ ~ ~ - - 1, a n d since Fo(Z ) is s u b o r d i n a t e t o G(z), t h e disk [z I < ~r 1 corresponds t o ~ subset o f t h e sheet R 5 b y F0(z), so t h a t F0(s) is u n i v a l e n t in [z[ < % / 2 - - - 1.
I t n o w follows f r o m K o e b e ' s t h e o r e m t h a t x > b l ( V ~ - - 1)/4. On t h e Other h a n d t h e inverse f u n c t i o n z = r o f Fo(Z ) m a p s t h e disk Iw[ < x i n t o t h e disk lzl < 1, so t h a t b y S c h w a r z ' s L e m m a bi -~ = ~b'(0) < x -i. T h u s we d e d u c e t h a t
4
x < b 1 < % / ~ - 1 x < 1 0 x . (3.2) T h u s Fo(z ) has t h e r e q u i r e d d e v e l o p m e n t (3.1), w h e n p = 1.
ON T H E VALIICON D E F I C I E N C I E S O F I N T E G R A L F U N C T I O N S O F I N F I N I T E O R D E R ] [ ~
W e n e x t n o t e t h a t No(z ) has u n b o u n d e d c h a r a c t e r i s t i c in Iz[ < 1. I n f a c t Fo(z ) c a n n o t h a v e a n y r a d i a l limits o t h e r t h a n p o i n t s o f E 0. I t follows f r o m a classical t h e o r e m o f F r o s t m a n a n d N e v a n l i r m a [5, p. 198] t h a t i f No(Z ) h a d b o u n d e d c h a r a c t e r i s t i c t h e n t h e t o t a l set of r a d i a l limits of No(z ) w o u l d h a v e a p o s i t i v e c a p a c i t y , giving a c o n t r a d i c t i o n . T h u s Fo(z ) has u n b o u n d e d c h a r a c t e r i s t i c .
This p r o v e s L e m m a 1 for t h e case p = 1. I f p > 1, we p r o c e e d as follows.
L e t E e be t h e set consisting o f all c o m p l e x n u m b e r s w p, such t h a t w E E . W e m a y s a y t h a t E e is t h e p - t h p o w e r o f E . L e t Fp(z) be d e f i n e d as a b o v e w i t h Ep instea~l of E , xP i n s t e a d of x, a n d set
1 1
F(z) = {Fe(zP)} ~- = b~(z ~ (b~zP+l[blp) -~ . . . ) .
Since Fp(z) ~ 0 for z ~ O, F(z) is regular. Also i f A o is t h e p a r t of {z L ~ 1 w h i c h c o r r e s p o n d s to t h e sheet R 5 b y Fp(z), t h e n Fe(z ) is u n i v a l e n t in A o. T h u s if Ap is t h e p ' t h r o o t of A 0, i.e. t h e set o f all z, such t h a t zp lies in A0, t h e n F(z) is u n i v a l e n t in Ap. I n f a c t i f zi, z~ lie in Ap a n d N(zl) ~ F(z2), we d e d u c e t h a t z~, z~ lie in Ao, Fv(z~) = Fp(z~), so t h a t zip = z~. T h u s we h a v e , for some i n t e g e r k, z~ = z 1 e x p (2uik/p). This implies F(zi) = F(z2) e x p (2uilc/p), so t h a t z 2 = zl, a n d F(z) is u n i v a l e n t in Ap, w h i c h includes t h e disk Izl ~ ( V ~ - - 1) l/p, a n d so c e r t a i n l y t h e disk Izl ~ % / 2 - - 1. W e also see t h a t if F(z) = w in E , t h e n Fp(zP) = we in Ep, a n d this is possible o n l y for zP in Ao, i.e. z in Ap, w h e r e F is u n i v a l e n t . T h u s Fp(z) assumes no v a l u e o f E m o r e t h a n once in Iz[ ~ 1. l~inally we see t h a t x P ~ b I ~ 10x p, so t h a t F(z) has t h e d e v e l o p m e n t (3.1).
T h e a b o v e a r g u m e n t assumes ~hat ~ does n o t m e e t t h e real axis, e x c e p t p e r h a p s a t t h e origin, t t o w e v e r , since Ep has c a p a c i t y a n d so linear m e a s u r e zero, E e will n o t m e e t e v e r y s t r a i g h t line t h r o u g h t h e origin, a t p o i n t s o t h e r t h a n w = 0. I f Ep does n o t m e e t arg z = ~, a -~ z, we a p p l y t h e a b o v e a r g u m e n t w i t h t h e set Ep(~) i n s t e a d o f Ep w h e r e Ep(~) is o b t a i n e d b y r o t a t i n g E v b y a n angle - - a r o u n d t h e origin. W e t h e n consider e~Fv(ze - ~ ) i n s t e a d o f F(z). T h e a r g u m e n t showing t h a t Fv(z) has u n b o u n d e d c h a r a c t e r i s t i c also shows t h a t F(z) has un- b o u n d e d c h a r a c t e r i s t i c a n d L e m m ~ 1 is p r o v e d .
W e can d e d u c e
],EMMA 2. Suppose that a D . . . , ap are preassigned complex numbers, not all zero, and let M = ~ = 1 la~i 9 Let E be the set of L e m m a 1. Then there exists Fe(z) regular in lzl ~ ], assuming no value of E more than 2p times there, having un- bounded characteristic in ]z I ~ 1 and a power series ~evelopment
N,(z) = alz + c~z ~ ~- . . . . apz" -~ O(z "+I) (3.3) near z : O. Further
IFp(z)l < IOM, for IzI <_ (VF2 -
1)/2. (3.4)
168 w . 1~. ttAYMAN
Suppose t h a t # > M a n d write
P av Zr --~- /.tZ 2P
~ o ( z ) =
t~ + ~ ~ z2p-~
T h e n [co(z)] = 1 for Iz] = 1, a n d og(z) has precisely 2p zeros a n d no poles in Iz] < 1 b y Rouch6's Theorem. L e t F(z) be t h e f u n c t i o n whose existence is asserted i n L e m m a 1, w i t h a 1 = / ~ , where M < / ~ < 10M a n d set
G(~) = F{o~(~)}.
We proceed to show t h a t _Fp(z) has the required properties. I t is evident t h a t
a n d s o
O)(Z) = [~--1 ~,P~=I av zv -~ O(ZP+I) ,
~-~p(Z) = ~t0)(Z) -~ 0 ( Z p + l )
has t h e required power series d e v e l o p m e n t a t the origin. ~Iext it is e v i d e n t f r o m t h e same a r g u m e n t concerning radial limits t h a t Fp(z) has u n b o u n d e d chaiacteristic in ]z] < 1. Also t h e e q u a t i o n ~o(z) = $ has precisely 2p roots in [zl < 1 for a n y ]~[ < 1, a n d so, since F ( z ) = w has a t m o s t one root z for w in E, it follows t h a t F ~ ( z ) = w has a t m o s t 2p roots for w in E.
F i n a l l y , since _F(z) is u n i v a l e n t in ]z] < r o = ~r it follows f r o m a classical i n e q u a l i t y for u n i v a l e n t f u n c t i o n [2, p. 4] t h a t IF(z)l < ttr~[z](ro --]z]) -2, ]z] < r 0. Also b y Schwarz's l e m m a l(o(z)l < ]z], for ]z I < 1, a n d so we deduce t h a t
Irp(z)l _< t~r~lco(z)](ro -- Ioo(z)[) -~ < aOMr~lzl(ro -- IzI) -2 < IOM, if Izl < ro12, This completes t h e p r o o f of L e m m a 2.
4. Proof of Theorem 2
W e shall proceed to construct t h e fimction of T h e o r e m 2
f(z) = ~,.~1 b,z", (4.1)
b y successively constructing its coefficients b,. W e set b 1 = 1, a n d assume t h a t pk is a strictly increasing sequence of positive integers such t h a t Pl = 1. W e assume t h a t b~ is k n o w n for n _< pk a n d proceed to construct bn for pk _< n _< Pk+l.
To do this, we shall i n d u c t i v e l y define a sequence ~ of positive numbers, increasing r a p i d l y to i n f i n i t y a n d such t h a t ~0 ~- 1. Suppose t h a t ~k has been
O N T t I E V A L I R O N D E F I C I E N C I E S O F I N T E G I ~ A L F U N C T I O N S O E I N F I N I T E O R D E R 169
chosen and l.et Ek be the set of Theorem 2. L e t Fk(z) be the function defined as in L e m m a 2 with p : l o k , E = E ~ and
an = b ~ . (4.2)
Then if a~ are the coefficients of F~(z) for all n, we cIefine bn by (4.2) for pk < n _~< ~%+~. I t remains to show t h a t the sequences ~k, Tk can be chosen in- ductively so tha~ f(z) given by (4.1) satisfies all the conditions of Theorem 2.
We start b y showing t h a t if ~ is chosen sufficiently large, when ~o~_1, ~o~ have been chosen, we shall have
b, < (2~_x) -~, s < ~ ~ i%+~ 9 (4.3) I n fact it follows from L e m m a 2, (3.4) a n d Cauchy's inequality t h a t
lanl < 1 0 ( { ~ / 2 - 1}/2) "M, P~ < n < p~+~, (4.4) where
Pk Pk
M = Ib, ie; < "Z Ib l.
v = l v ~ l
~u A o = (%/2--- 1)/2, for p a < n ~ p k + l we have
Bk = ~ = 1 Ib, l, we deduce from (4.2), (4.4) t h a t Pk
- - p - - n - - n
lbni < 10~k~ A0 Bk.
1 n
Thus (4.3) holds if ~-pk > 10(2~k_l/Ao)nBk, i.e. Qk > (10Bk)n-Pi(2Qk-1/Ao) n-Pi', and this condition is certainly satisfied for all n > pk, if
~k > lOBk(2ek-1/Ao) pk+l 9 (4.5)
I-Iere we use the fact t h a t Bk ~ ]bl[ = 1. We assume that, if Pk and ek-1 are known, ~ is chosen to satisfy (4.5) so t h a t (4.3) holds. Since ~--> co with k, we deduce at once t h a t f(z) given b y (4.1) is an integral function.
We note t h a t (4.3) implies in particular t h a t
Ibnl---<
1 for all n. Thus for~Iz[=~__<1/2, we have
If(z)l < ~ e" < ~ < 2e. (4.6)
- - ~ = 1 - - 1 - - e - -
L e t E~ consist of all points of E k other t h a n the origin, so t h a t by hypothesis E~
has a positive distance from the origin. We choose ~k to be positive decreasing, less t h a n half this distance and less t h a n 1/2. Then it follows from (4.6) t h a t f(z) assumes no value of E~ in ]z[ < ~k. Also for lz[ = e, where 0 < e < ~k, we have
~ ~ -- 2 ~
If(z)l ~ e -- e~ = ~ 1 1 - - - > 0 .
170 w . x . ~AY~A~r
Thus
f(z) # O,
for 0 < lz] < ~k, a n d so in this a n n u l u sf(z)
assumes no value of Ek. Alsof(z)
has a simple zero a t t h e origin.The f u n c t i o n
Fk(z/~k)
b y our construction a p p r o x i m a t e s v e r y closely tof(z)
a n d the coefficients of b o t h functions are t h e same, n a m e l y a~, for n _< Pk+l.
Also for n > ~ok+~ we h a v e in view of (4.3)
Ib.I < ( 2 Q k ) - = 9
This will enable us to show t h a t
f(z)
a n dFk(z/ek)
behave similarly for ]zI = rk < ~k, provided t h a t p~+~ is large enough. H o w e v e r before constructing rk a n d Pk+~we n e e d some f u r t h e r conditions on eL, which like (4.5) will be satisfied i f eL is sufficiently large. Accordingly we choose ek so large t h a t in a d d i t i o n to (4.5) we h a v e
2p~ < ~1(1 e~), (4.7)
a n d
2p~ log < q~l( } ~k) log (1 ~k) 9 (4.8)
W e n o w suppose t h a t ~ satisfies (4.5), (4.7) a n d (4.8), a n d proceed to define rk.
I t follows f r o m L e m m a 2 t h a t
Fk(z/~k)
has u n b o u n d e d characteristic in [z[ < ok.Thus we m a y choose rk, such t h a t
1 (4.9)
- f f ~ k < r k < Qk, a n d in a d d i t i o n
T{rk, Fk(Z/Qk)} > qi2(O~) + 1 . (4.10)
N e x t we n o t e t h a t sets of c a p a c i t y zero h a v e linear measure zero, a n d hence so do their inverse images b y regular functions, since the inverse f u n c t i o n is locally con- formal except a t isolated points. I n p a r t i c u l a r the inverse image of Ek b y
Fk(z/ok)
meets [z[ -- r o n l y for a set of r of linear measure zero. Thus, b y increasing rk if necessary, we suppose in a d d i t i o n to (4.9) a n d (4.10) t h a t
Fk(z/ok)
does n o t m e e t Ek, for [zl = rk. Since Ek is compact, this implies t h e existence of a q u a n t i t y ek, such t h a t 0 < s k < 1 a n diFk(z/Q~) -- al > sk,
for a e E L a n d ]z[ : rk. (4.11) I-Iaving chosen rk to satisfy (4.9) to (4.11), we proceed to show t h a t if Pk+l, which has so far been left u n d e t e r m i n e d , is chosen suitably, t h e n (2,1), (2.2) a n d (2.3) will be satisfied.We write
Fk(z/~k)
=- ~,n~=lB,~z",
a n d n o t e t h a t t h e series is absolutely convergent for ]z I = rk. Thus we m a y choose Pk+l so large t h a t ~n~=pk+l [B n ] [z] n < I sk, [z] ---- rk, where s~ is t h e q u a n t i t y in (4.11). N e x t it follows f r o m (4.3) a n d the fact t h a t r k < ~ k , t h a t for [z[ = r ~O ~ T H E VALIt~OI~ D E F I C I E N C I E S O F I1NTEGI:tAX~ F U I ~ ( ] T I O S I S O~ I l q ~ I I ~ I T E OI~DEI~ 1 7 l
~,L~k+l+l Ib-llz[ ~ ~< X~n~176 2-n---- 2--Pk+l '
a n d this is less t h a n 89 sk if Pk+~ is large enough, which we assume. T h u s we m a y choose Pk+~ so large t h a t (regardless of a n y later choices of e~, r~, a n d p~
for ~ > k - k l ) we h a v e
, I ~ . ( 4 . 1 2 )
If(z) --
F,(z/~k)I
= i~pk_l_l_kl (bn - -B,,)z~l
< ek, lz] = rkI t follows f r o m this a n d (4.11) t h a t for a in E~ t h e equations
f ( z ) = a
a n d
Fk(z/Qk) = a
h a v e e q u a l l y m a n y roots in Izl < r~, i.e. at m o s t 2pk, in view of L e m m a 2. N o w (2.1) follows at once f r o m (4.7), (4.9) a n d t h e fact t h a t r increases.l~lext if
n(t,a)
denotes t h e n u m b e r of zeros off ( z ) - - a
in 0 < [z t < t , it follows f r o m t h e d e f i n i t i o n of ~k, " t h a t for a E Ekn(t, a) = O, t < ~k , while from w h a t we h a v e j u s t shown
T h u s i f a # 0
n(t, a) < 2pk,
(Sk < t < rk.rk
N(rk, a) = f n(t, ta)dt
< 2j0k log (rk//~e) .0
I f a = 0, we recall t h a t , since
f(z)
has a simple zero a t t h e origin, a n d no zeros in 0 < [ z l < ( ~r k
= --f n(t, O) --dt q_
log ~k ~ 2pk log(r~/(~k) . N(rk, O)
d t
~k
T h u s for a in Ek we h a v e in all cases
N(r~, a) < 2~o~ log
(r~/~)
< ~1(I e~) log (1 e~) < ~l(r~) log r~,in view of (4.8), (4.9) a n d t h e fact t h a t r increases w i t h t. This proves (2.2).
F i n a l l y we h a v e b y (4.10), a n d the well-known i n e q u a l i t y for t h e characteristic of the s u m of t w o functions [5, p. 162],
q~(r,) + 1 < r + 1 < T{rk, Fk(z/ek)} <_ T{rk, Fk(z/ek)
-- f(z)} +T{r,,f(z)} + 1
= T{r~,f(~)} + 1,
in view of (4.12) a n d t h e fact t h a t e~ < 1. This proves (2.3) a n d completes t h e p r o o f of Theorem 2.
172 w . K . HAY~AN
References
1. AHLFORS, L., E i n Satz y o n Henri Cartan uncl seine A n w e n d u n g a u f die Theorie cler mero- morphen l~unktionen. Soc. Sei.-fenn. comment, phys.-math. 5, :No. 16 (1931).
2. HAV~IAN, W. K., Multivalent functions (Cambridge University Press 1958).
3. --~-- Research problems in Function Theory (Athlone Press of the U n i v e r s i t y of London 1967).
4. HYLLE~COI~E~r A., Valiron deficient values for functions meromorphie in the plane. Acta Math. 124 (1970), 1--8.
5. ~EVANLI~NA, l=~., Eindeutige analytische Funktionen (Springer, Berlin 1936).
Received March 29, 1972 W. K. ]-Iayman
Imperial College of Science a n d Technology Dept. of mathematics
Exhibition Road London S.W. 7
