Value distributions of entire functions in
regions of small growth
DAVID DRASlX a)
1. Statement of results
L e t f(z) be a n e n t i r e f u n c t i o n of f i n i t e o r d e r O. I t is classical (of. [2, Ch. 4];
[6, Ch. 1] t h a t a p r o x i m a t e order o(r) m a y be associated w i t h f(z) so t h a t t h e c o r r e s p o n d i n g indicator f u n c t i o n
log [f(re~~
h(O) = lim sup rO(r ) (0 < 0 < 2~) r-~oo
is continuous, 2n-periodic, a n d t r i g o n o m e t r i c a l l y convex. L e t I = (a, fi) be a n o p e n i n t e r v a l with
h(O) <_0 ~ < 0 < l , (1.1)
a n d choose 00,~ < 00 < I. W e s a y t h a t t h e c o m p l e x n u m b e r a is m a x i m a l l y a s s u m e d n e a r { a r g z = 0 o } if t h e r e is some e > 0 such t h a t for all d > 0
n(r, a, 0o, d)
lim sup re(r ) _> e; (1.2)
r-+co
here n(r, a, 0 o, d) d e n o t e s t h e n u m b e r o f r o o t s o f f ( z ) - - a, including m u l t i p l i c i t y , in t h e region {]z] < r } f l { [ a r g z - 00l < ~}. T h e set of all m a x i m a l l y a s s u m e d v a l u e s n e a r {arg z = 0o} for a given e > 0 will be d e n o t e d b y ~(00, s).
More generally, for a closed s u b i n t e r v a l 11 = [~l, i l l of I , let n(r, a, 11) d e n o t e t h e n u m b e r of r o o t s o f f(z) - - a, including m u l t i p l i c i t y , in t h e region
{1~1
< r} n {~X 1 < a r g z < ill},1) Research partially supported by (U.S.) National Science Foundation. The author also thanks the Royal Institute of Technology, Stockhohn, for its hospitality and support during much of the preparation of this paper.
2 8 2 D A V I D D R A S I l g
and set
~ ( I 1 , e) = {a; lim sup
r --> oo
N o t e t h a t ~(I1, s) D U~,<o<~, ~ ( 0 , e).
n(r, a, I1) ]
r~(e ) ~ s/" (1.3)
T~EOR}~M 1A. Let 11 =
[gl, ill]
be a closed subinterval of I = (~, fi) where (1.1) is satisfied. Then there exists a positive sequence {a=},r;~+ 1/r ---> oo, (1.4)
and a sequence {a~} of complex numbers with the property that i f w E ~(11, e), then
lw - - a.I < e - ~ (1.5)
for infinitely m a n y n.
THEO~EN lB. Let a sequence {a,} of complex numbers be given along with a positive sequence {a,} satisfying (1.4), and let
= ['1 ~ {w; lw - - a~ I < e-~ (1.6)
r n ~ . O n>_ m
Then there exists an entire function of f i n i t e order whose indicator vanishes on an interval I = (~, ~), and such that for some 0 o C (~, fl) and some s ~ 0
~ ( 0 o , ~) ~ ~ . (1.7)
2. Remarks
The indicator h(O) is non-negative on a set which includes an interval of length ~/~, so t h e h y p o t h e s i s (1.1) requires ~ > 89 Since ~(r) - + ~, it is no loss o f generality to suppose
1 < ~(r) < 2 ~ (r ~ 0 ) . (2.1) The examples of T h e o r e m 1B h a v e order ~ for a n y ~ C (89 1), with e in (1.7) equal to z-1 sin ~. B y Considering f ( z '~) (n = 2, 3 , . . . ) we o b t a i n examples for all orders ~ > 1, Q # 1, a n d a more intricate construction, which we do n o t give here, yields functions of order I which satisfy (1.7) for some ~ > 0. There is p r o b a b l y a relation b e t w e e n the largest s allowed in (1.7) and the variables ~ a n d (fl - - ~).
I n [7, p. 55], G. Valiron asserted t h a t ~(00, e), for a f i x e d e > 0, can n e v e r be as large as t h e c o m p l e m e n t of a single p o i n t with respect to the finite plane (i.e., {arg z = 00} cannot be a Borel direction of f(z)); as far as I a m aware, he n e v e r published a proof. Since ~ e -~ < o% it follows from T h e o r e m 1A t h a t U~>0~(00, e) has (planar) measure zero.
V A L U E : D I S T R I B U T I O N S O F E N T I R E : F U N C T I O N S I N : R E G I O N S O:F S M A L L G R O W T H 283 T h e c h a r a c t e r i z a t i o n s of ~ ( 0 0, e) a n d ~ ( I 1, s) g i v e n here i n v i t e c o m p a r i s o n w i t h t h e r e c e n t s t u d y o f A. H y l l e n g r e n [4] on V a l i r o n deficiencies o f m e r o m o r p h i c functions of f i n i t e order. H y l l e n g r e n s h o w e d t h a t if f is m e r o m o r p h i c a n d o f finite order, a n d if A[s] = {a; A(a) >_ e}, w h e r e A(a) is t h e Valiron d e f i c i e n c y o f t h e c o m p l e x n u m b e r a, t h e n hie] is c o n t a i n e d in a set of t h e f o r m (1.6) w h e r e t h e a~ s a t i s f y an+l/a~ = 0(1), r a t h e r t h a n (1.4). T h u s , t h e c o n s i d e r a b l y smaller sets
~ ( I 1, e) are also o f c a p a c i t y zero a n d h a v e H a u s d o r f f m e a s u r e zero for all m e a s u r e f u n c t i o n s h(t) such t h a t
f h(t)(-- log t)-~t-~dt < oo.
0
(I t h a n k P r o f . H y l l e n g r e n for several discussions on these m a t t e r s ) .
i n 89 t h a t T h e o r e m 1A is false w h e n 11 T h e f u n c t i o n e ~ , ~ - ~ 2 , f i = shows
is r e p l a c e d b y I .
Notations. A c o n s t a n t w h i c h d e p e n d s o n l y on e (of (1.2)), fi - - ~, fil - - c% or
~(r) (where ~(r) is s u b j e c t to (2.1)) will be g i v e n w i t h o u t r e f e r e n c e to t h e s e q u a n - tities. Most inequalities are v a l i d o n l y for s u f f i c i e n t l y large r = EzE, a n d such a n i n e q u a l i t y will be qualified b y r > r o or r > r0(K); in t h e l a t t e r case, r o d e p e n d s on K as well as ~(r), fil - - ~1, fi - - ~ or s. A n y o f t h e s e expressions will be f r e e l y used t o d e n o t e d i f f e r e n t c o n s t a n t s in d i f f e r e n t c o n t e x t s .
3. Proof of T h e o r e m 1A
W e f i r s t n e e d a P r o p o s i t i o n w h i c h allows (1.3) t o be r e p l a c e d b y a m o r e con- v e n i e n t condition.
PROPOSITliOlg 1. For c~ C ~ ~'~(Ii, e), let
{ n ( r ' a ' I 1 ) }
_R(a) = r; rQ(r ) < 3e/4 . (3.1)
Then there exists M ~ > 1 and r 1 = rl(a ) such that
n(r, a, 11) -- n(r',
a,/1)
>ler~(T) (r E ~(a),
rl(a )< r' <_ r/M~~
(3.2) LEM~A 1. With e as in (1.2), there exist ro, M o with(r/Mo)~~176
< 4-1~r e(r) (r > r0). (3.3) Proof. Choose M 0 so t h a t for some $ > 0,Mol/2e ~ < 4-1e; (3.4)
2 8 4 DAVID DI%ASIN
there is no h a r m in supposing ~: so small t h a t log M 0 < 1.
Now ~ ' ( t ) t l o g t - + O as t - ~ o % so there is rl(~ ) with [~'(t)tlog t] < 89 (t > r~(~));
further there is r o ( > rl(~)) so t h a t
1
lo 0/
log Mo]"
Then if 3 / o l r > %, (3.5)--(3.7) yield t h a t
{ l ~ 1 7 6 }
]~(r)-- ~(r/Mo) [ ~ a2~Zlog 1 ~- l o g r ~ l o g 3 / ~
~2 log Mo(log r) -1 ~ ~(log r) -1, so (2.1), (3.4) and (3.8) lead to
(r/3/o)e(r/M~ r q(O : Moe(~/iO)re(r/Mo)--~(r) < Mol/2e~ < 4-1e which is (3.3).
(r > ro),
(3.5)
(3.6)
(3.7)
(3.8)
LEMMA 2. There exists ro(a ) with
n(r, a, I1) ~ 2(2r) e(2r) (r > to(a)). (3.9) Proof. This is an immediate consequence of Jensen's theorem [2, p. 9], the defining inequality log M(r) < {1 + o(1)}r e(r) and
2r
log 2 _< n(r, a) l o g 2 < f n ( t , a ) t - l d t < N ( 2 r , a) (r > 1).
n ( r ~ a , I1)
r
I t is now easy to obtain Proposition 1. L e m m a 1 (with 89 0 in place of 3/0) and L e m m a 2 imply t h a t there are M0, ro(a ) with
n(r/3/o, a, 11) <_ 4(log 2)-1(2r/3/o) Q(2r/M~) ~ er e(r) (r > to(a)) ,
and the Proposition, with 3 / ~ = Mo, follows from this and the obvious inequality n(r, a, I 1 ) - - n(r', a, I1) ~ n(r, a, I1) -- n ( r / M o, a, 11).
It is also useful to have a slight sharpening of (1.1). According to (1.]), there exists r -+ 0 (r -+ oe) with
m a x log ]f(re~~ < r ~(') (r > 0). (3.10)
V A L U E D I S T t t I B U T I O : N S O F E N T I R E g U N C T I O : N S I ~ I ~ E G I O N S O F S M A L L G ~ O W T ] : ~ 2 8 5
(cf. [6], p. 71). F o r K > 1 consider t h e closed regions D ( K , r ) a n d D I ( K , r ) given b y
D ( K , r) : {te~~ r / 2 K < t < 2Kr, or < 0 < fl}, (3.11) D~(K, r) = {te~~ r / K < t < Kr, ~ ~_ 0 ~_ fi~}. (3.12) Since t h e f u n c t i o n (r'/r)z m a p s D ( K , r) onto D ( K , r') a n d DI(K, r) onto D I ( K , r') it follows t h a t there is a positive c o n s t a n t ~(M) w i t h the p r o p e r t y t h a t
inf GD(K,~)(Z, b) = ~(K) (z, b e DI(K, r), r > 0) (3.13) where GD(K, ~)(Z, b) is t h e Green's f u n c t i o n for D ( K , r) w i t h pole a t b.
LEMMA 3. There exists an increasing unbounded function K(r) (r > O) such that, if s is the constant of (1.2) and T is given by (3.13)
m a x log [f(~)[ < ls~(K(r))r~(r) r > ro(r (3.14)
E D(K(r), r)
and, further,
Proof. L e t K 1
o f
~(K(r))r ~(0 = o{7:(K(s))s e(s)} (r, s -+ ~ , s / r - + ~).
(3.15)
= 4 a n d for j = 2, 3 , . . . d e t e r m i n e K i as t h e largest solution
~(Ki) ~ 2-1/4T(Ki_l), (3.16)
gj_l < gj <_ 2gj_l. (3.17)
Since T(K) is a continuous f u n c t i o n of K , it follows t h a t K i exists a n d K i -->
as j--> ~ . I f rl(j) is chosen so large t h a t
[e(t) -- e(r)] ~ log 2(log r) -1 (rl(j) ~ r / K s < t < Ksr) (3.18) (this is possible, as can be seen from t h e p r o o f of (3.8) in L e m m a 1), t h e n (2.1) a n d simple m a n i p u l a t i o n s give
r . e(r) ~_ r ~ 2r (r~(j) ~_ r/Kj < t < Kjr) Since r --> 0, we n o w h a v e an ro(j, r ( > rl(j) ) w i t h t h e p r o p e r t y t h a t
r ~(') _~ 2-13/as~(Kj)r ~(') (ro(j, p) ~_ r / K i < t < Kjr). (3.19) L e t us f u r t h e r require t h a t ro( j + 1, r ~ K~.ro(j, r a n d let K(r) = K i w h e n Kjro(j, r < r < Kj+~ro( j + 1, r I t is easy to see from (3.10), (3.16) a n d (3.19) t h u t (3.14) holds as well ~s
K(r)-Xr --> oo (r --> co). (3.20)
To complete t h e p r o o f of L e m m a 3, we show (3.15). Suppose 16r < t < 32r,
286 D A V I D D R A S I N
with r so large t h a t (3.18) is satisfied with Kj > 32. Then (3.16) and (3.17) imply t h a t "r > 2-~/~(K(r)) and this, (2.1) and (3.18) lead to
~(K (t) )t ~(0
~(K(r))re(r) > 2-1/q61/2re(0-e(r) ~ 2 ~/4 (r~(K3) ~ t/32 < r < t/16), (3.21) and iteration of (3.21) easily gives (3.15).
Finally, we can prove Theorem 1A. Let a C ~(I1, e) and let R(a) be as in (3.1).
Let r*(a) be so large that, with M + as in (3.2), K(r) > M ~ if r > r*(a) and log + la[ _~ -~sT(K(r))r ~(r) (r*(a) ~ K(r)-lr) ; (3.22) (3.15) and (3.20) show t h a t r*(a) exists. We write R*(a) for R(a) FI (r*(a), ~ ) . Then if r C R * ( a ) , z E D l ( r ) and {b,} are the roots of f - - a in D l ( K ( r ) , r ) , we have from Poisson's formula ([2], p. 7)
log [f(z) -- a I < f log [f(~) -- alK(~, z)d~ -- ~, G(z, b~)
~eOD (3.23)
(Z 6. DI(K(r), r), r 6. R*(a)).
Here K > O,
f
z)d = 1. Then (3.14) and (3.22) showlog If(C)
- a[ ~ -~ez(K(r))r ~(~) (~ 6. aDl(K(r), r), r 6. R*(a)), and since {b~} are in DI(K(r), r), (3.2) and (3.13) imply t h a t>_
Thus
log If(z) -- a I ~ ~e~(K(r))r "(~) -- a(r, a) (z 6. D(K(r), r), r 6. R*(a)). (3.24) Hence if a' 6. ~(I1, e), and [a' -- a I > ~ ' r ( K ( r ) ) r ~(~), it follows t h a t
R(a') fi (K(r)-lr, K(r)r) = 0 if r 6. R(a).
Thus let {t~}--> ~ so slowly t h a t
t,~+l/t,~ < i n f K ( t ) (K(t,~_~)-~t,,_~ < t < K(tm+~)tm+ f i m : 1, 2, . . . ) and let J ~ = [t~, t~+~]. First let ms be the least positive integer with
J~, FI {U R*(a); a
6. -~(11, e)}
:~ ~,and choose r ~ 6. J~,, a~, 6. ~(I~, e) with rm, 6.R*(a,~). Then let m2 be the least positive integer > ml with
J,,~ fl {(K(r~)rm,, oo)} lq {U R*(a); a 6. ~(I~, e)} ee O,
and choose r ~ 6. J ~ , a ~ 6. ~(I~, e) with rm~ 6. R * ( a , ~ ) . . . This gives sequences {r~}, {a%} which we label simply as {r.}, {a.}, and let
~ V A L U E D I S T I ~ I B U T I O I q S O:F : E I ~ T I I ~ E ~ U I ~ C T I O N S I:N" ~ E G I O N S O F SlYIA]bL G I ~ O W T I : s 287
r = l s~(K(r~))r ~ (3.25)
Since t h e {r,} increase, a n d r~+l/r,_l ~ K(r,_l), we h a v e t h a t r,+l/r~--)-oo a n d so, f r o m (3.15), ~ . + 1 / ~ , - ~ oo w h i c h i s (1.7). F i n a l l y , let a C ~ ( I 1 , s), a n d s E R*(a).
Then s belongs to some i n t e r v a l J ~ a n d t h e construction given guarantees t h a t there is a n rp (p ---- m -- 1 or m) which belongs to t h e sequence {r,} w i t h either 1 ~ s / r e ~ K ( r e ) or 1 ~ s / r p ~ K ( r ~ ) -~. T h e n (3.24) a n d (3.25) w i t h a = a~, r = r~ ensure t h a t [a -- ae[ ~ e-"~, a n d T h e o r e m I A is established.
let T h e n
L e t {a,}
4. Proof of Theorem 1B
be a sequence for which ~ + 1 / ~ - - > ~ , and, for a f i x e d ~ E (89 1),
~, z -- 9(cos ~ ) r ~ -- log 2 (n ~ 1, 2, . .). (4.1)
r,+l/r, --> ~ , (4.2)
a n d (4.1) yields a relation b e t w e e n r , a n d ~ which we keep for t h e r e m a i n d e r of this p~per. Given ~ or r~, which satisfy (1.4) or (4.2), there is no loss o f g e n e r M i t y in decreasing t h e ratios ~,+~/~, or r,+l/r . so t h a t also
(log ~,_1) 6
(log s,+l) 2 - ~ ~ " (4.3)
W e m a y t h e n s t a t e Theorem 1B more precisely as
TI~EOgEN 1B'. For ~ < ~ < 1, let {(~,} be a sequence which satisfies (1.4) and (4.3), and define {r,} by (4.1); f i n a l l y let {a.} be a sequence with
. [(log r._l) 6 }
[a,l < m m ~(~gog r,+l) ~' 89 r,_l) 6_ . (4.4) Then there exists an entire function f(z) with
h(O! ~ 0 Further, we have for all ~ ~ O,
log M ( r , f ) ~ r ~ ( r - ~ ~ )
and, i f h(O) is the indicator of f(z) with respect to o(r) -~ ~, (larg z -- ~l < 89 -- ~/2p)).
in the notation of (1.2), that n(r,, w, ~, ~)
Jim i n f r~(r,) ~ ~-1 sin ~
n --> o9
(4.5)
(4.6)
(4.7)
2 8 8 ]:)AVID DI%ASII~
for all w C N,~ [.Jm>, Cn, where
C . = { w ; Iw - a . I < e - ~ " } . ( 4 . 8 )
The function f(z) is obtained b y Riemann surface methods, and depends on the existence of an auxiliary entire function g(z) which satisfies Theorem 1B' with all a, identically zero. We list the requisite properties of g(z) below in Proposition 2, and then show how to modify g to obtain f. In w 5 is a proof of Proposition 2.
PROPOSITION 2. There exists an entire function g(z) which satisfies (4.5) and (4.6). Further, i f {r,} is the sequence which appears in Theorem 1B', there exist sequences {R,} with I~,/r,---~ ~ and r,+l/R,---> ~ , and {~,}--~0 such that
n(r, w, ~, (~)
inf ~ (l -- ~.)~-~ sin z~ (4.9)
rn/2 ~_ r ~ 2r n r ~
for all w satisfying
]wl ~ 2e -~'.
Finally, we can choose en---> 0 so slowly that e,R~ > (log Rn) 7
with that property that i f
D . = {R._I < I~l < R.} n {~ ~ larg ~I > ~/4},
and E . = OD., then
log [g(z) l ~ e._l(R._a) ~ (n > no; z C E.).
(4.10)
(4.11)
(4.12)
(4.13) We accept this Proposition for now, and produce f(z) using an indirect approach.
Using g(z), we shall construct a continuous function F(z) which is regular in the complement of certain simply-connected resgions
{A,.,n} ( n = 1, 2 . . . . ; m = 1 . . . . , ~ ( n ) )
with Am,,G D , for all m and n, where D , is defined in (4.12). Inside the {A~.n}, F will not be holomorphic, b u t will be nearly so in the following sense:
each A~, n can be divided into three subregions in each of which F(z) = F(x, y) = u(x, y) + iv(x, y) has continuous partial derivatives, and
IF~/F,] ~ A(log [z]) -2 (a.e. z E/I,,,,~) (4.14) for some positive constant A, where, as usual
V A L U E D I S T R I B U T I O N S OF E N T I R E F U S T C T I O ~ S I N I:r OF S M A L L GROWT1K 2 8 9
1 i
F, = ~ (u~ + vy) + ~ (v~ -- u~),
1 i
F~ = -~ (u~ - vz) + ~- (v~ + u~).
(4.15)
a n d so
N > t
ff
Finally, we will show t h a t
(0 < ) p(z) -- 1 _< A(log [z[) -2, (a.c.) (4.16) dxdy
{ p ( ~ ) _ ~} ~ < ~ . (4.17)
F maps the plane topologically onto a R i e m a n n surface 7. (4.18) The u t i l i t y of (4.17) a n d (4.18) arises from results of O. Teichmuller a n d 1 ).
Belinskii ([5, Ch. 5, w 6]). F o r these conditions i m p l y t h a t ~ is parabolic and, if fl(~) maps t h e C-plane conformally onto 7, t h e n for a suitable choice of A, t h e
induced t r a n s f o r m a t i o n $(z) = A lf{l(F(z)) satisfies
~(z) ~ z ( z ~ ~ ) . (4.19)
A l t h o u g h F is n o t regular, we h a v e maxl~l= r IF(z)[ ~ r ~, a n d this a n d (4.18) allow t h e expressions h(O) a n d n(r, a, 0o, (~) to be defined for F(z) as if F were entire. Our explicit construction of F will g u a r a n t e e t h a t (4.9) is satisfied for those w which belong to i n f i n i t e l y m a n y of t h e discs (4.8) so t h a t (4.19) yields t h a t
f(z)
= f l ( A z ) meets all conditions o f T h e o r e m 1B'.Thus we 'start w i t h g(z), as in Proposition 2, a n d for z E D , describe how to achieve the F(z) which will satisfy (4.17) a n d (4.18). L e t
3. = (log Rn_l) 6
a n d consider t h e closed subsets A.~.. o f D . in which ig(z) E ~ 3n (Z C ~m,n)
Note t h a t (4.11), (4.12), (4.13) a n d (4.21) i m p l y t h a t Am,,, we m a y consider n fixed in this construction. I f an Mobius t r a n s f o r m a t i o n
W - - a n L w = e i p n 3 2 n 2
3 n - - d n W
which m a p s t h e disc {[w[ < 3.} to itself, w i t h pn Then L induces a m a p s so t h a t
(4.20)
(4.21) D n for all m. Thus satisfies (4.4), consider the
(4.22) chosen so t h a t L(v,,) = 3n.
This will i m p l y t h a t t h e d i l a t a t i o n p(z) of F (cf. [3, p. 439], [5, p. 18]) satisfies
2 9 0 b A W D DIIASIN
for ~11
L ( r , e ~(~ = The ~~ s(0) = 0. (4.23)
0, a n d we can now define t h e m a p p i n g H from {]w I _< ~ ] to itself as
l o g v ~ - - 1 0 g 2 _ < U < l o g v n , 0 < V < 2 z . Since exp {} is conformal, we h a v e t h a t
H(w)~/H(w)w = k ( W ) u w,
a n d we can c o m p u t e t h e left side of (4.29) using (4.25), (4.23), (4.28). Thus (4.22), (4.23) a n d (4.26) show t h a t I s ' ( V ) - 11 <_Ala,/'c~l,
Is(V) -- VI <_ 2 ~ A [ a , / ~ l . I t is t h e n easy to show t h a t
for
o < ~ < _ 1 ~
}1 (4.24)
log u -- log z./2 89 < u < T.
log 2 -- --
F(z) = H o L o g(z) (z 6 Am,~) (4.25)
where L is specified in (4.21). F o r z ~ U~ ,dm, n we set F(z) = g(z); it t h e n follows from (4.22)--(4.25) t h a t F is continuous in the fuII plane.
The n e x t t a s k is to show t h a t F satisfies (4.7) a n d (4.8). Consider t h e disc {jw -- a,J < e-"n}. Since R , > rn, (4.4) a n d (4.20) i m p l y t h a t this disc is inside {]w[ _< 1Tn}. I t t h u s follows f r o m (4.22), (4.3), (4.4), a n d (4.20) a n d t h e interlacing of t h e {R~}, {r,} t h a t there is a c o n s t a n t A (independent of n) w i t h
an (log r~_l) 6 1
I1 - - I L ' ( w ) ] ] _~ A - - ~n < A (log Rn_~) 6 (log rn+~) ~ ~< A(log %+1) -2 (4.26) which t e n d s to zero as n - ~ ~ . N o w L(0) = a,, so (4.26) implies t h a t i f n is sufficiently large, t h e inverse of { [ w - - a ~ ] < e -~n} u n d e r L is c o n t a i n e d in {lzl < 2e-"~}, a n d (4.7) follows from (4.9), (4.10) a n d (4.19).
I t remMns b u t to v e r i f y (4.17) (or (4.16)) a n d (4.18). E v i d e n t l y F~ = 0 i f z ~ Um.n A ... a n d t h e r e p r e s e n t a t i o n (4.25) shows t h a t it suffices to show
(H(w))~/H(w)), ~ A(log [z[) -2 (w --- g(z), z 6 A~,~); (4.27) f u r t h e r since g is a regular, t h e explicit f o r m u l a (4.24) shows we need o n l y consider 1 < [wl < T ~ } along these z for which ~ 1 < ]g(z)l _ < ~ . W e cut A = { w ; ~ n _ _
t h e axis {arg w = 0} a n d write ue iv = exp (U Jr iV). T h e n (4.4) m a y be w r i t t e n H(ue~ ~) = exp {k(U @ i V ) } = exp { K ( U ~ i V ) ~- i K * ( U -~ iV)} w i t h
K ( U + i v ) = u
(u - log To/2) (4.2S)
K * ( U @ i V ) : V ~- (s(V) -- V) log 2
(4.29)
(4.26) a n d a n d
H(~@iv) ~- ( u exi) [~ {v _~ (8(v) _ v)
a n d define F(z) for z 6 A .... b y
V A L U E D I S T R I B U T I O N S OF E N T I R E F U N C T I O N S I N B, E G I O N S OF S M A L L G R O W T H 291
K u ~- 1 ,
K v = O,
so t h a t , for perhaps a different c o n s t a n t A
]F~/F~I < Ala./~.l
a n d t h u s (cf. (4.26))
] K * - - 1] < ls'(V) -- Ii <
Ala~/v,[
IK*[ ~ I s ( V ) - Vt ~_ 2xA]a~/v~I,
(z r Am,,3
IF~/F~I ~
A(log r~+l) -2 < A(log ]z]) -~(z E Am,~);
since D~ c {]z I < r~+l} a n d this proves (4.16). To o b t a i n (4.18), we observe t h a t t h e image of Am, ~ b y g is a bordered R i e m a n n surface, u n d hence so is t h e image o f A~,, u n d e r F . F is also regular in the c o m p l e m e n t of t h e d ... ~nd since F is u n i q u e l y d e f i n e d on aA ... (4.18) follows from s t a n d a r d gluing a r g u m e n t s (cf.
[1, pp. 117--119]).
5. Proof of Proposition 2
The m e t h o d s used here rely h e a v i l y on Chapters 1 a n d 2 o f [6].
Suppose
go(z)
is a canonical p r o d u c t of order 0, 89 < Q < 1 w i t h go(0) | 0, a n d let {b=} be t h e roots of go. M a n y functions can p l a y t h e role of go below, b u t all will have, for some absolute c o n s t a n t K ,n(r, O) < K r Q
(5.1)(K m a y be t a k e n as 6, for example). L e t r 0 > O, A > 0 be given, a n d define p r o d u c t s =l(Z) a n d n2(z) b y
~x(Z) : ] - [ (1 --
z/b,~);
~2(z) = " ~ (1 --z/b,).
(5.2)ibm] < A-% Ibm/> A~, o
The discussion of [6, pp. 62--3] a n d (5.1) i m p l y t h a t , given s I > 0, there exists
Ao(sx)
(which also depends on t h e absolute c o n s t a n t K of (5.1)) such t h a t ifA ~__ Ao(Sl)
[log[zl(Z)I[ + ]log[z~(z)[ I < si re (5.3)
if
re A-1 < ]z I < roA.
(5.4)One f u r t h e r element of flexibility will be needed. L e t M be a (large) positive integer a n d let {am(0)} (m = 0, ~: 1 . . . . :~ M) be a f a m i l y of 2a-periodic trigono- metrically convex functions of order Q, 1 < 0 < 1. Thus each h~ is continuous, has r i g h t a n d l e f t - h a n d derivatives which agree off a n a t m o s t countable set of 0, a n d
292 DAVID DRASIN
o
8=(0) = < ( 0 ) - (0 < 0 < 2=) (5.5)
0
increases (in (5.5), h'm d e n o t e s e i t h e r t h e r i g h t or l e f t - h a n d d e r i v a t i v e o f ha).
I n o u r s i t u a t i o n , sin(O) will increase o n l y b y simple j u m p s a t one or t h r e e values of 0, a n d t h e r e exists a set E ( M ) = {0 o, 01, 0_ 1, 02, 0_8} outside of w h i c h all func- tions h,~(O) are c o n t i n u o u s l y differentiable. To m e a s u r e t h e denseness of t h e f a m i l y
{ k i n } l e t
q ( M ) = m a x m a x [h'm+a(O) - - h'm(0)]. (5.6) --M<_m<_M--1 o~E(M)
T h e n for each m, C h a p t e r 2 o f [6] yields an e n t i r e f u n c t i o n fm whose i n d i c a t o r is hm(O). T h i s f ~ has several p r o p e r t i e s w h i c h are useful here a n d so we i n d i c a t e t h e salient f e a t u r e s o f t h e c o n s t r u c t i o n . F o r 0 < 0 < 2~, let
Am(O) = (2~0) -1 lim {h'(O -~ 6) - - h'(O - - 6)} (5.7)
a + 0
m e a s u r e t h e j u m p of t h e d e r i v a t i v e of h~ a t 0, a n d o b s e r v e f r o m our c o n v e n t i o n t h a t A,,(O)----0 for all 0 ~ E ( M ) . T h e n for j - - - 2 , . . . , 2 we place ni, m(r) zeros of f ( z ) on {arg z = 07} to satisfy
Ini,,,(r) - - A,,(Oj)rQI < 1; (5.8)
f,,(z) is t h e canonical p r o d u c t whose zeros are so d i s t r i b u t e d . T h e n , to each 61 ~ 0
t t
is a PM w i t h t h e p r o p e r t y t h a t if Iz] = r > p M
r~hm(O) - - e~r Q < log ]fm(z) l < r~h=(O) q- sir ~ ( - - M < m <_ M ) (5.9) save for p o i n t s z c o n t a i n e d in circles C~,k whose r a d i i rm, k (k = 1, 2, . . .) satisfy"
r - l ~ rm, k < Sl ( - - M < m < M , r > p'M) (5.10) (the s y m b o l ~ m e a n s s u m m a t i o n o v e r t h o s e k such t h a t C~,k i n t e r s e c t s {[z[ < r}). Also, we o b t a i n f r o m (5.6), (5.7) a n d (5.8) t h a t g i v e n e~ > 0, t h e r e exists a q l > 0 a n d PM tt such t h a t if q ( M ) < q l , t h e n
Inj,=(r) - - ni.m+l(r)] < e2r Q r > PM. (5.11)
t t!
F i n a l l y , we let PM ~-- m a x (PM, PM)"
F o r N = 1 , 2 , . . . , let q ( N ) = N -2 a n d t h e n consider a f a m i l y of 2 M q - 1 t r i g o n o m e t r i c a l l y c o n v e x f u n c t i o n s hm(O) where t h e specific choice of M will be m a d e later. E a s i e s t t o define is
ho(O)
= cos 50 (101 _< ~);t h e r e m a i n i n g f u n c t i o n s are d i v i d e d into t w o classes, each of M functions. T h o s e in Class I will be labelled h i , . . . , hM a n d we f i r s t describe these. Choose
01, 0 < 01 < ~g -- ~r/:/2~ with
V A L U E D I S T I I , I B U T I O N S O F E N T I R E F U N C T I O N S I N R E G I O N S O F S M A L L G R O W T t I 293
and, in t h e i n t e r v a l define
COS ~01 = . N - 1 ( 5 . 1 2 )
0 < 0 < 01, let h,~(O) = ho(O) for 1 _< m < M. N e x t , we
hM(z~ ) --~ (2N) -1 (5.13)
a n d t h e n , for 1 _<m < M ,
m
hm(~) = h0(~) + ~-(hM(~) - - h0(~)). (5.14)
F o r 01 < 0 < 3, h~ is tile u n i q u e p a r t i o n of a sinusoid o f p e r i o d 2~/~ which a t ;~ a n d 01 i n t e r p o l a t e s t h e v a l u e s hm(n) a n d ho(O 0 (to see h o w this sinusoid is c o n s t r u c t e d , of. [6], p. 52; u n i q u e n e s s follows since ~ - - 01 < ;~Q-1). N e x t , for
< 0 < 2~ let hm(O) = h,n(2= - - 0). T h u s , in t h e e n u m e r a t i o n o f E ( M ) , 01 t h e solution of (5.12), 0_ 1 - - 01 a n d 00 = =. T h e f u n c t i o n s in Class I I are w r i t t e n h 1 .. . . , h_M, a n d are c o n s t r u c t e d as in (5.12), (5.13) a n d (5.14) save t h a t m is r e p l a c e d b y - - m , N b y N 4. 1 a n d 01 b y 02, where 02 is d e f i n e d b y t h e e q u a t i o n cos ~0e = (N 4. 1) -1. N o t e t h a t t h e f u n c t i o n s h,n(O) are 2z-periodic a n d trigono- m e t r i c a l l y convex. T h e easiest w a y to establish this c o n v e x i t y is to v e r i f y t h a t each s,,(O) (defined in (5.5)) increases. To see t h a t S,n increases, we o b s e r v e t h a t h,~(O) is a c o n t i n u o u s f u n c t i o n a n d is sinusoidal at all points of c o n t i n u i t y of h'm; a t t h e r e m a i n i n g p o i n t s of t h e d o m a i n h'm has a positive j u m p d i s c o n t i n u i t y .
W e can n o w r e l a t e t h e choice of M to N a n d t h e sequence {r~} which is specified in t h e s t a t e m e n t of P r o p o s i t i o n 2. Choose {t.} w i t h
rn/t,_l = t,/r, (5.15)
so t h a t b o t h sides o f (5.15) t e n d to i n f i n i t y as n - - ~ oo. W i t h ex(N ) = N -2 as m e n t i o n e d a b o v e , in (5.3), (5.9) a n d (5.10), choose A ----AN so large t h a t (5.3) holds w i t h e I ---- sI(N ) a n d t h e n choose PM, M ( M = M ( N ) ) a n d e 2 ( = s2(N)) so t h a t i f t h e {fro} are chosen as in (5.8), t h e n (5.11) m a y be s h a r p e n e d to
[nj, m(A~r) - - % m_~(A~vr)] < r'~ log AN) -2 (r > PM). (5.16) A c c o r d i n g t o (5.7) a n d (5.8), (5.16) can be a c h i e v e d b y m a k i n g [A~(0) - - Am_a(O)]
small for all 0, a n d these differences will be d i m i n i s h e d if q(M) is small, i.e. i f M is large.
W e n e x t choose n ( N ) so large t h a t n ( N ) > n ( N -- 1),
log (t,+~/t~) > 4 ( 2 M ( N ) @ 1) log AM(N) (n > n ( N ) ) (5.17) a n d in a d d i t i o n , w i t h PM(N) selected so t h a t (5.9) a n d (5.10) hold w i t h o u r choice o f el, we also h a v e
r~ > PM(N) (n > n(N)). (5.18)
F o r each n , n ( N ) <_ n < n ( N 4- 1), t h e i n t e r v a l (t., t,+l) is d i v i d e d into
294 DAVID DRASIN
( 2 M ( N ) - } - 1 ) intervals (aj(n),/~j(n)) w i t h
cr ) =ocj(n)/fij(n)
for all i a n dj (-- M(N) <_ i, j < M(N)).
W h e n t h e value of n is clear from t h e context, we a b b r e v i a t e~2(n)
a n d fij(n) b y ~j a n dfij.
We set, for each n, T / ( =Tj(n))=
{z;aj_< Izl < f i j } , a n d for t h e m o m e n t suppose n # n ( N + 1 ) - - 1. T h e n in Tj, g is assigned t h e same zeros as t h e corresponding fi i f j > 0, a n d as f_j i f j < 0; if n = n ( N 4 - 1 ) - - 1, t h e n in T j g has t h e same zeros as f_j for all j .
(This special definition, w h e n n =
n(N +
1 ) - 1, allows a s m o o t h connection near {[z L = t,,(N+l)}). F i n a l l y in {[z[< t l } , g
is assigned the same zeros asfMO)"
W i t h {b,,} these zeros, we set
g(z)
= ]-[(1z/bn).
The p o i n t of this construction is t h a t if z E Tj a n d
fj(z)
is t h e proper choice of 2~ orf_j,
as explained above, t h e nlog Ig(z)l = log [fj(z)(z)l +/~j(z) (z e Tj) (5.19) where, for large n,
I/~/(z)l < 2N-3zelzl ~ (5.20)
outside circles Ck of radius rk such t h a t
r-1 ~ r rk < el(N) = o(N -1) (n(_/Y) < n <
n(N +
1)) (5.21) (cf. (5.10)). G r a n t i n g this for t h e m o m e n t , it is easy to complete t h e p r o o f of Proposi- tion 2. Indeed, (5.21) implies there exist {R~} --> oo w i t hR~/t, --~ 1 (n --> co)
such t h a t (5.19) a n d (5.20) hold on all of {[z[ = R~}. I n particular, this, (5.9) a n d t h e fact t h a t]h~M(O)l ~
( 2 N -~ 2) -1 (0 < 0 < 27C) i m p l y for large n t h a tlog
[g(R~e~~ >_
R~{(2N + 2) -~ -- N -2 -- 25" 3/2} >_ (3N)-~R~(5.22) (0 < 0 < 2 ~ , n > n 0 )
On t h e r a y s { a r g z = s we have
log
[g(re~='~/4)[
> 89 ~:~)r e (r > r0) (5.23) Sinceh,,(O) >_
cos (1:~) for all 0 w i t h 0 < 0 < ~z, (5.23) is clear from (5.9) a n d (5.19) if z does n o t belong to t h e circles e s t i m a t e d in (5.10); if z is interior to one of these circles, t h e n it follows from (5.8), (5.19) a n d (5.20) t h a tf(z)
does n o t v a n i s h in t h e circle, a n d so (5.23) follows from (5.9), (5.19), (5.20) a n d t h e m i n i m u m principle. Thus (5.22) a n d (5.23) i m p l y t h a t (4.13) holds w i t h a n y e, >_ (4N)71 (n > n o,n(N) < n <_ n(N -[-
1)), so (4.11) can be achieved as well, b y increasing t h e n u m b e r sn(N)
i f necessary.Similar reasoning gives (4.5) a n d (4.6). Indeed, w h e n (5.20) is valid, these con- clusions follow from t h e construction of t h e
h~(O)
since m a x oh~(O)
= 1, a n d t h e i n e q u a l i t yh~(O) <_ 2N -1
w h e n ]0 -- zl < ~/2~ a n d --M(N) <_ m < M(N).
F i n a l l y , we consider (4.9) a n d (4.10), a n d let
{s,}
be a sequence w i t h 89 _< s~ < 2r.. T h e n s~ is well-contained inTo(n)
in t h e sense t h a ts~/c~o(n) -~ oo
as n--> oo. L e t { ~ } ~ 0 a n d {A~}-> o% {S~}-~ oo be sequences (withV A L U E D I S T I ~ I B U T I O N S 0:~ E N W I ~ E I ~ U h V C T I O I ~ S :~1~ ~ E G I O : N - S O F S ~ A L L G ~ O W ~ r H 295
cr < Am < S,~ < s,~, A,, ~ a 0 ( n ) a n d S . ~ s ~ as n--> c~) so t h a t (5.19), w i t h m = O, holds on all of {Iz] = A.},{[z[ = S.} a n d t h e segments
{arg z = ~ ~ On, A~ ~
lzl _< S.}.
T h e n if D* d e n o t e s t h a t region b o u n d e d b y these curves w h i c h contains a s e g m e n t of t h e n e g a t i v e axis, our c o n s t r u c t i o n implies t h a t g has at least
k(S,,) --~ ~-1 sin ~ (S~ - - A~) - - 2 ~ ~-1 sin ~ SnQ(1 d- o(1))
zeros in D* w h e r e t h e r a t e a t which o(1) t e n d s t o zero d e p e n d s on n b u t n o t t h e choice o f s, E [~r,, 2r~]; f u r t h e r , (5.9), w i t h m ~-- 0, a n d (5.19) y i e l d t h a t
log [g(~)] ~ log I f 0 ( ~ ) [ - [/~0(~)] >~ 2 c o s ~ 0 I~l ~ ~ o(1)[~i ~~
> 3 c o s s p [~l ~ > 3 c o s ~ ( 3 r , ) ~ ~ 9 c o s ~ r ~ (~C O D * , n ~ n o ) . H e n c e , b y R o u c h d ' s t h e o r e m g(z) assumes e v e r y v a l u e w w i t h
]w] ~ e x p (9 cos ~p r~) (5.24)
a t l e a s t k(S~) t i m e s for z E D~*.
F o r a f i x e d (3 ~ 0 a n d all large n, if w satisfies (5.24)
n(s~, w, ~, 6) - - n ( R , , w, ~, 6) > n(S~, w, 7~, ~ ) -- n ( A , , w, z~, ~,)
(5.23)
> ~-1 sin 7~ e s~:(1 ~- o(1)),
a n d so (4.9) a n d (4.10) are consequences of (5.22), (5.23) a n d t h e d e f i n i t i o n (4.1).
W e conclude b y s k e t c h i n g a p r o o f o f (5.19) a n d (5.20) p r o v i d e d z avoids t h e circles e s t i m a t e d b y (5.21). L e t z0 E Tj, [z I --~ r0 and, for c o n v e n i e n c e of n o t a t i o n , suppose - - M ( N ) d- 1 <_ j ~ j(Zo) <_ M ( N ) - - 1. T h e n t h e i n t e r v a l (A~:ro, A~ro) m e e t s a t m o s t one T~ (k ~ j ) (cf. (5.17)). L e t {b,} a n d {b,,j} d e n o t e r e s p e c t i v e l y t h e zeros of g(z) a n d f~(z), a n d g i v e n a sequence {a~}, define ~ * ( 1 - z/a,~) t o be t h e p r o d u c t o v e r those n w i t h r A ~ ~ < Ia~l < roA ~.
Since e~(iv) = iv'~, (5.2) gives
]log lg(z)l - - log ]fj.(z)]] _< [log]z*(l - - z/b~,j) 1 -- log Iz*(1 - - z/b,)ll d- 2iv-st ~. (5.26) H o w e v e r , t h e (b~} a n d (b,.]} agree in Tj, a n d t h u s (5.I6) implies t h a t
[log I7~*(1 - - z/b.,j)[ - - log [~*(1 - - z/bn)[[ = ]log ] ~ I(1 - z / a . ) ~ . [ I , (5.27) where s~ = :[: 1, a n d p(zo) <_ (iv log AN)-2r ~. L e t
p(zo)
Q~o(Z)
= I I ( 1 -z/a~
n = l
I t is clear t h a t if Q1 is a n y p a r t i a l p r o d u c t o f Q=o, t h e n
_ A 2 v~(Nlog A~)-~} ~ 3N_2(log Alv)_lrO -
]og IQi(z)l < ]og {(1 -~- ~T/
(iv > No) (5.2s)
296 D A V I D D R A S I : N
and Cartan's estimate ([6], p. 21 with
2eR
= 12r) and the manipulations leading to (5.27) ensure t h a t outside circles Ck whose radii satisfyt h a t
~ rk <_ 6r(NA~) -1
(5.29)log [Ql(Z)[ > - - [2 ~ - l o g
(12eNA~)]
m a x log [QI(~)I]r = 1 2 r
(5.30)
-- 3[2 @ log
(12eNA~)]N-2(log A~v)-~r ~.
F r o m (5.28) and (5.30), it is clear t h a t (5.27) is e s t i m a t e d b y
llog l~*(1 --
z/bn, j)l
-- log [~*(1 --z/b,)l[ <_ N-a/2r~
(r > ro), (5.31) and (5.26) and (5.31) give (5.20). Finally, the b o u n d (5.21) is a direct consequence of (5.29).References
1. A ~ F o ~ s , L., SARIO, L., Riemann Surfaces, Princeton, 1960.
2. CA~W~mKT, M., Integral Functions, Cambridge, 1956.
3. GOLDBERG, A., OSTI~OWSKI, I. V., The distribution of values of meromorphic functions (in l~ussian), Nauka, Moscow, 1970.
4. ttYLLE~GR~X, A., Valiron deficient values for meromorphic functions in the plane, Acta Mathematica 124 (1970), pp. 1--7.
5. LEHTO~ O., VIRTANEN, K. I., Quasikonforme Abbildungen, Springer, 1965.
6. L E v ~ , B., Distribution of zeros of entire functions, A.3/I.S. Translation, Vol. 5, 1964.
7. VAYLIRO~, G., Directions de Borel des fonctions m6romorphes Memorial des sci. math. 89 (1938).
.Received May 14, 1973 David Drasin
D e p a r t m e n t of Mathematics P u r d u e University
W . Lafayette Indiana 47907 U.S.A.
