The asymptotic behavior of functions extremal for Baernstein’s cos Βλ theorem

T H E A S Y M P T O T I C B E H A V I O R O F F U N C T I O N S E X T R E M A L F O R B A E R N S T E I N ' S cos/3A T H E O R E M

By

JOHN ROSSI AND JACK WILLIAMSON

1. I n t r o d u c t i o n

The classical Wiman-Valiron cos 7rA theorem [2, chapter 3] states that if f is an

entire, non-constant function of order A( / )= A < 1, then

log m lim *(r) => cos ~'A , ~ log M(r)

(1.1)

where

M ( r ) = sup lf(re'~ m.(r) = inf I[(rei~

A generalization of this theorem due to Kjellberg [13], [14] removed the

condition on the order of fi He proved that for any p with 0 < p < 1 either there

exists arbitrarily large r with

(1.2)

or else

(1.3)

log m , (r) > cos 7rp log M(r)

log M(r) _ !im rp - a (0 < a =< + o:).

His theorem implies that if tt (/) = / z < 1 is the lower order of f, then

- - log m . ( r ) > (1.4) !im ~oogM-~ = cos "rrtt.

It also implies that if A ( f ) < 1 and

(1.5) log m , ( r ) = < cos 7rh log M(r) (r _--> ro),

then (by (1.3) with p = A) f grows regularly in the sense that ~ (f) = A (/).

Drasin and Shea [4] investigated entire functions of order A < 1 extremal for the

cos 7rA theorem in the sense that f satisfies (1.5) asymptotically; that is,

(1.6) log m,(r)-<_ (cos 7rA +o(1))logM(r) (r--~).

128 JOURNAL D'ANALYSE MATHI~MATIQUE, VoL 42 (1982/83)

ASYMPTOTIC B E H A V I O R OF FUNCTIONS 129

For such functions, they were able to determine the exact growth of logM(r) ,

log m , ( r ) and N(r) for "most" r (where N(r) is the usual Nevanlinna integrated

zeros functional). In fact, they found that (1.6) was enough to obtain an asymptotic

determination of log lf(re'~ for "most" r and all 101 -< zr. Their results imply that f must locally resemble a Lindel6f function.

A remarkable generalization of Kjellberg's result was obtained by Baernstein [1].

He proved

T h e o r e m A. Let f be an entire, non-constant [unction. Let [3 and p be numbers

satisfying

(1.7) O < p < o o , O < f l _ < ~ -, f lp<zr.

Then either there exist arbitrarily large r such that the set {O:loglf(re '~ cos [3p log M(r)} contains an interval o[ length at least 2[3 or

logM(r) (1.8) !ira r p

If we define

(1.9) m o ( r ) = s u p ( in[ loglf(re"~ r \ Io l~ /

and if [3 and p satisfy (1.7), then Theorem A can be simply stated as

(1.10) ma(r)<cos[3p logM(r) (r > ro) ~ (1.8).

Baernstein asked whether the behavior of functions "extremal" for his theorem

can be characterized in much the same way that Drasin and Shea characterized

functions extremal for the cos ~rA theorem (see [1, p. 182]). This paper answers this question. We shall say that an entire [unction o[ order A (0 < ;t < oo) is extremal [or [3

if

(1.11) m~ (r) < (cos/3A + o (1))log M(r) (r --> oo)

where [3 satisfies (1.7) with p = A. I[ A = O, we say that f is extremal [or all [3 ~ (0, rr]. Notice that if /3 = 7r and A < 1, (1.11) reduces to (1.6) and an asymptotic

determination of log If(re'~ I is possible as mentioned above. For other values of [3

and A we can roughly summarize our results as follows: If [ is an entire function

which is extremal for [3, then

(I) log M(r) and ma (r) display very "regular" growth for "most" r, and

(II) there are "very long" sectors of opening "nearly" 2[3 in which the

asymptotic behavior of log If(re'~ can be precisely described and in which [ has

"very few" zeros. As we will explain below, if [3 < 7r, the behavior of log [[(re'~ described in (II)

130 J. ROSSI A N D J. W I L L I A M S O N

cannot in general be improved unless more is assumed of f. In this direction we

have (III) if f has only negative zeros, is of non-integral order, and is extremal for/3,

then log lf(re'~ displays very "regular" growth for t01< ~r and "most" r.

2. S t a t e m e n t a n d d i s c u s s i o n o f r e s u l t s

In order to state precisely the results mentioned above we need to recall some

familiar terminology. A measurable subset G of [1, ~) has logarithmic density one if

f c t-~dt = 1. (2.1) !im log r nI,,rj

It has density zero if

(2.2) lim ] dt = O. r ~ J G n i l , r ]

Further, G is very long if G has logarithmic density one and

(2.3) G = U [ a . , b . ] where a , . - - -~ , b, . /a . , -+~ as m--+r rt'l ~ 1

Also recall that a positive function L(r ) varies slowly on G provided that given

any A > 1

L(crr) !irr~ L ( r ) = 1 r E G

holds uniformly for A -1 =< o- =< A.

Our first theorem gives a precise meaning to result (I) of the Introduction.

T h e o r e m 1. Let f be an entire function of order A (0~ A < ~ ) , which is

extremal for ~. Then there exists a very long set G and a set H of density zero such that

(2.4) log M ( r ) = r AL(r)

where L varies slowly on G and

(2.5) m~ (r) = (cos/3,~ + o (1))r~L (r) (r ~ ~, r E G - H).

We remark that (2.4) and (2.5) imply that

m~(r)=(cost~A +o(1 ) ) l ogM(r ) ( r - - ~ % r E G - H ) ,

a result which Ess6n [8] has already proved using different methods. We further

remark that since f has order A, (2.4)implies that f has regular growth; that is,

(2.6) lim log log M (r, f ) = h. , ~ log r

ASYMPTOTIC BEHAVIOR OF FUNCTIONS 131

This is implicit in the work of Drasin and Shea [4]; a detailed proof may be found in

[t6]. Finally, we note that results of Drasin [3] and Hayman [10] show that the set G

in Theorem 1 is "best possible." They construct functions with only negative zeros

satisfying (1.11) for/3 = 7r such that L(r) does not vary slowly on (0,oo).

We can replace the order h of f by its lower order /x in Theorem 1 but only by

allowing G to have upper log density one (i.e., G satisfies (2.1) when lim,~= is

replaced by limr~=). In this case we write log dens G = 1. Precisely, we have

T h e o r e m l a . Let f be an entire function of lower order tz (0 <- ix < ~) and

suppose that (1.1l) holds with h replaced by tz. Then there exists a set G satisfying

(2.3) with log dens G = 1 and a set H of density zero such that

(2.7) log M(r) = r 'L (r)

where L varies slowly on G and

(2.8) rn~ (r) = (cos fllx + o(1))r~L(r) (r --~ % r E G - H).

Because the proof of this theorem is obtained by a suitable modification of the

corresponding result for the cos 7tO theorem found in w of [4], it will be omitted.

The interested reader may consult [16] for the complete details. Our next theorems make precise results (II) and (III) of the Introduction. Before

stating them, we need to introduce some notation. We will denote by {r,,},

r,, E (a,,, b~), any sequence in the very long set G of Theorem 1 for which

(2.9)

For cr > 1, we let

r,,/a,~,b,~/rm-->~ as m --> ~.

(2.10) /,~(o-)=[o--~r,~,o-r,,], A(cr)= U I.,(cr). tt 'l = 1

Once the sequence (r,,} is fixed, we denote by {r'} and (r"} any two sequences in G

such that

{ a m < r ' < r , , < , " < b m , m = 1 , 2 , . - .

(2.11) r,./r,.,r,./r,.-->oo' " as m--~Qo.

tV J For convenience assume r , , < ra+~, m = 1,2, . �9 �9 , and set

(2.12) I,~ = [ r ' , r~'], A = U I,~. m = l

We also introduce the sectors

(2.13) ~,.(o';oJ, a ) = { r e " : r E I , . ( c r ) , l O - o ~ l < a }

and

132 J. ROSSI AND J. WILLIAMSON

(2.14) ~,. (to, a ) = {re'" : r E L , IO - to I< a}.

As usual, ~ . ( o - ; to, a ) and 207,(to, a ) will denote the closure of these sectors, and

n (S) will denote the number of zeros of f (counting multiplicity) which fall into the

bounded set S. If S = {z : iz 1---- r}, we write n(S) = n(r).

We can now make precise result (II) of the Introduction.

T h e o r e m 2. Let f be an entire function of order A which is extremal for fl and

let G be the very long set of Theorem 1. Let r,, be any sequence in G satisfying (2.9)

and let {tom} be any sequence satisfying M(rm) = [f(rme"- )[. Then if 0 < ~ < [3 and

~ > 1

(2.15) n ( ~ 7,(o-; to,,, or)) = o(log M(r,,)) (m --.~o)

and

(2.16) log l fire'")[ = [cos(0 - to,,)a + o(I)]logM(r)

uniformly in r and 0 as r ~ ~, re'9 E ~ 7,(tr; to,, a ) - ~,., where ~,, is the union of a

finite number of closed disks the sum of whose radii = o(r,.) as m ---~oo.

By (2.16) extremality for [3 implies very specific behavior in the sectors ~ 7,(or). However, outside these sectors the behavior of logl[[ cannot in general be

determined. Indeed, if A > �89 and 0 </3 --<_ ~r/2A, it is easy to construct functions which are extremal for [3. By a theorem of Levin and Pfluger [15, chapter 2, theorem 3] one need only construct a subtrigonometric indicator function

h(O)=t cos~ 1o1<--/3 tcos/3X,

corresponding to the proximate order p(r )= A. By constructing other indicator functions which agree with h(O) for l 01 =< [3, it can be shown that many different types of behavior can occur outside the sectors 9 , , when [ is extremal for [3 (see [16] for details, and compare (2.18) below). This makes this extremal problem somewhat different in character froro the extremal problem for the spread relation solved by Edrei and Fuchs [6], [7, especially pp. 142, 143].

For non-integral A > 1 and ~r/2A </3 < ~rlA, we cannot in general construct

functions extremal for [3. (Some specific examples where [3 depends on A are given in [16].) Baernstein has asked in private correspondence whether there exists a non-integral )t > 1 and a/3, ~r/2A </3 < 7r/)t, such that no f of order )t is extremal

for [3. We are unable to answer this question. On the other hand, if the zeros of [ are distributed on a ray, then extremality for

[3 implies a complete asymptotic determination of f. Our next theorem thus makes precise result (III) of the Introduction.


T h e o r e m 3. Let f be an entire function of non-integral order A with only negative zeros which is extremal for [3, and let G be the very long set of Theorem 1. Then for any sequence {rm} in G satisfying (2.9) and for any o- > 1 there exists a set A(tr) of the form (2.10) such that

(2.17)

Moreover

(2.18)

n(r)= [ tsin TrA [ + r~L(r) (r---> ~, r ~ A(cr)).

log lf(re'~ = ( - 1) q [cos Oh + o (1)1 r~L (r) (r E A(cr))

uniformly in r and O, ] 01 <= ~r - ~ < rf, as r ---> ~ in A(o'), where q is the genus off.

We note that Theorem 3 implies that f is locally a Lindel6f function. Conversely

we note that every Lindel6f function of non-integral order A is extremal for some [3,

[3;t <~-.

3. P r o o f o f T h e o r e m I

The key ingredients to the proof of Theorem 1 are a subharmonic version of a

theorem of Drasin and Shea and a variant of Baernstein's *-function used by him in

proving the cos [3A theorem [I]. In our proof of Theorem I we will assume that

A > 0 , since Drasin and Shea have shown [4] that if A = 0 then l o g M ( r ) = L(r) where L(r) grows slowly on a set G as in Theorem 1 and

l ogm. ( r ) - -L ( r ) ( r - - ~ % r E G - H , densH=O).

Consider a function u (z) which is subharmonic in the plane C (for background

material on such functions the reader may consult [11]). Define

M(r, u) = sup u(re'~ m.(r, u) = inf u(re'~

1 N(r, u)=~--~ ~_,, u(re'~

We can now state the subharmonic version of a theorem of Drasin and Shea [4]

which will be needed in the proof of Theorem 1.

T h e o r e m B. Let u(z ) be subharmonic in C of order #, 0 <= p < 1. If

(3.1) m.(r,u)<=(cos~rp+o(1))M(r,u) (r--->oo),

then there exists a very long set G and a set H of density zero such that

(3.2) M(r, u) = rPL(r)

where L varies slowly on G,

134

(3.3)

and

(3.4)

J. ROSSI A N D J. W I L L I A M S O N

m,(r ,u)=(cosTrp+o(1))r~L(r) ( r - - ~ o ~ , r E G - H )

N(r ,u )=[s inrrP+o(1)] r~L(r ) ( r - -~o%r~G). L rrp j

In addition to Theorem B, Theorem 1 also depends on the properties of

Baernstein's *-function. Following Baernstein we define the function v (re'~ by

(3.5) v (re ,0) = sup f log If(re '~ )[ d& 1

where the sup is taken over all intervals of length 20. In [1] it is proved that

(a) v (z) is subharmonic in Im z > 0 and continuous on Im z > 0 except perhaps

at z = O; (b) for each /3 E(O, Tr], v(re '~) is a nondecreasing convex function of logr

(0 < r < oo);

(c) for each re'~ there exists an interval I of length 20 for which the sup in (3.5) is

attained. For /3 E (0, 7r], let I~ (r) be an interval of length 2/3 such that

v(re'~) = f l~ Z~(r)

Define

(3.6) /x~ (r) = inf{log If(re'*)l: 4, ~ Is (r)}.

Now assume A (f) = A and that/3 satisfies (1.7) with A in place of p. By definition

of tn~ we have

(3.7) /z~ (r) =< rn~ (r) (0 < r < ~).

In proving Theorem 1 there is no loss of generality in assuming f(0) = 1, since A > 0

and hence f is not a polynomial. Thus

lim v(re i~) = 0. r ~ . O +

Let H(re '~) be harmonic in 0 < argz </3 with boundary values

(3.8) H(r) = O, H(re '~) = u (re '~).

(Since/3A/or < 1 and the order of v(re '~) is at most A, these boundary values do in

fact determine a harmonic function.) For Im z _-> 0 define

(3.9) h(z ) = H ( z ~) (y =/3/7r).


h (z) is harmonic in the upper half plane 0 < arg z < 7r with boundary values

h(r) = O, h(re '~) = h ( - r) = v(r'e'~).

As in [1], we obtain, for 0 N 0 < 7r,

f r sin 0 (3.10) h(re'~ = 1 v(t 'e '~) r 2 t 2 dt rr + 2tr cos 0 +

o

and

i,o (0.) (3.11) h~176 1 + dv,(t) ho - ~ , o

where v,( t)= yt 'v ' ( t 'e '~ (By the log convexity of v(re'~), v~ is a positive Borel

measure.)

It is easy to see from (3.11) and the definition of v~ that ho may be extended in an

obvious way to a function subharmonic in C with order p =/3A/Tr < 1 and that

(3.12) M(r, ho) = ho(r),

(3.13) m .(r, he) = h o ( - r),

(3.14) N(r, ho) = 1 v(r,e,~)" ~r

((3.14) follows from the subharmonic version of Jensen's theorem.)

Baernstein proved the following relationship between ho, Ho and f [1, p. 192]:

ho(r)=yHo(r ' )>=2TlogM(r* ) ( 0 < r < ~ ) (3.15)

and

(3.16)

h o ( - r)+ ho(r) = y[Ho(re~'~)+ Ho(r~)]

P r o o f o f T h e o r e m 1.

~p =/3A, we have

--- 2 T [/x~ (r ~) + log M(r ~)]

By (1.11), (3.7), (3.15),

(0<r<~).

(3.16), and the fact that

ho ( - r ) + ho (r)-< 2T [1 + cos 7rp + o (1)]log M(r +)

(3.17) <=[l +cos Trp + o(1)]ho(r) (r---~).

Hence, by (3.12) and (3.13) we have

(3.18) m.(r , ho)<=(cosTrp+o(1))M(r, ho) (r---~).

Thus ho satisfies the conditions of Theorem B and so there exists a very long set G~

such that

136 J. ROSSI A N D J. W I L L I A M S O N

(3.19) ho (r) = rPL,(r)

where L~ varies slowly on GI,

(3.20) ho ( - r) = (cos 7rp + o(1))rPLjr)

and

(3.21) N(r, h o ) = l v ( r ' e ' ~ ) = [sin f ro+ o(1)] ho(r) 'n" L "n'p

By (3.19) and (3.20) we have

(3.22) ho ( - r) + ho (r) = [1 + cos rrp + o (1)]h0 (r)

Comparing this to (3.17), we deduce that

1 (3.23) log M(r ~) = ~ (1 + o (1))ho (r)

Then (3.7), (3.16), (3.22), and (3.23) imply

(r ~ 0% r (E G1 - H, , dens H, = 0)

(r ~ ~, r E G,).

(r---> oo, r E G 1 - H i ) .

(r--~oo, r E G,- -Hi) .

1 (3.28) L ( r ) = ~ (1 + o(1))L, ( r ' / ' )

and L(r) varies slowly on O.

(3.27)

(3.25) implies

1 log M(r) = ~ (1 + o (1))ho (r l'v)

(3.25) = l---(l+o(1))r~LJr'/~) (r---~oo, r E G - H )

23,

and

(3.26) m~(r)~-(cosflA +o(1))logM(r) (r--->oo, r E G - H ) .

Since Ll(r 1/~) varies slowly on G and d e n s H = 0, (3.25) is valid for r E G. Thus if

we set

L (r) = r-* log M(r) (r > O)

(r E G)

(3.19), (3.23), and (3.24) imply

Since

p =AA =3"A,

(3.24) 2Tm~(r')>-(cosTrp+o(1))ho(r) (r-->oo, r ~ G l - H ~ ) .

Now define G = {r : r ~/~ E G1}, H = {r : r ~/~ E Hi} and note that

log dens G = log dens G1 -- 1 and dens H = dens H~ = 0.


Further, (1.11) and (3.26) imply

(3.29) m~(r)=(cosl3A +o(1))logM(r) (r---~oo, r ~ G - H ) .

]?heorem 1 now follows from (3.27) and (3.29).

Also by (3.19), (3.21) and the definitions of G and L we obtain the following

equation to be used in proving Theorem 2:

(3.30) v(re'~)=[2sin~x A+o(1)] r~L(r ) (r---~,rEG,.

4. Proof of T h e o r e m 2: P r e l i m i n a r y resul ts

The proof of Theorem 2 is rather lengthy and will be divided into six main

sections. In w we obtain local asymptotic estimates for the harmonic function H

introduced in the proof of Theorem 1. Because v is subharmonic with the same

boundary values as H, we will be able to obtain locally an asymptotic upper bound

for v.

In w we use a variant of Petrenko's formula due to Edrei and Fuchs to obtain

(2.15), while in w we obtain local approximations to v by functions v,, which are

integrals of log [/I over fixed intervals of length 20. In w we exploit the fact that jf

has very few zeros in the sectors @m of (2.15) to construct functions ~b,, which are

harmonic in @,, and approximate log bfl except on small exceptional disks. In w we

use some methods of Baernstein to prove (2.16) for ~b,~ (and hence for log l[I) assuming a technical lemma proved in w

We begin the proof of Theorem 2 by obtaining an asymptotic estimate of the

function H used in the proof of Theorem 1 and assume as we did in that theorem

that [ (0)= 1. Let {rm} be any sequence in G satisfying (2.9). We shall keep this

sequence fixed throughout the remainder of the proof of Theorem 2. If the function

L(r) and the set G are as in Theorem 1, then by an appropriate "diagonalization" (see e.g. [6, p. 70 or p. 90]) we obtain sequences {r'} and {r"} satisfying (2.11) and

intervals Im of the form (2.12) such that if r EIm

(4.1) L(r)=(l+o(1))L(rm) (r---~o%r~I,~).

Equations (2.4) and (4.1) now imply that if r E Ira,

(4.2) log M(r) = (1 + o(1))rXL(rm) (r ---~o% r E Ira).

In view of (3.8), (3.24) and the definition of Im we obtain

(4.3) H(re'~)=[2sin~A A+o(1)] r~L(r,, ) (r--~,rElm).

Further, since h(z)=H(z ~) for Imz =>0 and h0 attains its maximum on the

positive real axis ((3.12)), it follows from (3.25) and (4.2) that

Ho(re'~ (r---~,rEI,,,O<=O<=[3).


Since H ( r ) = 0, an integrat ion yields

(4.4) 0 -< n ( r e ' ~ <- 7trio(r) = (27 + o(1) ) r~L(r" ) (r--->o~, r E In, 0<= 0 <-_ ~).

From (4.3), (4.4), the fact that H ( r ) = 0 and /3A < 7r, and the s t ructure (2.11) and

(2.12) of the I,,, we obta in the fol lowing l emma.

L e m m a 4 . 1 .

H ( r e ' ~ ( r - - > ~ , r E I m , O<-O<=[3)

where o(1)--->0 uniformly in O.

(We r e m a r k that the Im in L e m m a 4.1 may be p roper ly con ta ined in the I,~ of

(4.3) but the s t ructure (2.11) and (2.12) remains intact. This abuse of nota t ion will

cause no confusion.)

P r o o f . By applying the conformal t r ans fo rmat ion z---> z ~/~ we may assume

/3 = 7r and A < 1 with no loss of general i ty . Define

G" (re,O) = H ( r e ,O)_ (2 s A O_._____AA) r~L(r" )

where re '~ E ~ = {re '~ : r ~ Ira, 0 < 0 < ~-}. We see tha t G " is ha rmon ic in ~m

while on 0a//" we have, by (4.3) and (4.4),

I G " ( r e , - ) l < = e,.r L(r , , ) ,

O" (r) = O, (4.5)

I G " ( r ' e ' ~ ) t <= K ( r ' f L ( r m ) , 0 < 0 < ~,

= K~r" 'XL" , IGr,(r"e '~ < ~,~) trm), 0 < 0 < ~ ' ,

where em = e " ( r ) - - - > 0 as m--->~ un i formly for r E I " and K is a constant

i ndependen t of rn and 0. vv 10 < Let Wm (re'O) be the ha rmonic m e a s u r e of the semicircle z = r"e , 0 = 0 <= 7r with

respect to the semidisk I z I =< r'm', 0 --< 0 =< It. W e have (see e.g. [5])

[ r " - re'~ wm (re'~ = _ 2 arg " r '.--;~-~--~"

7r \ , . * r e /

< tt It is easily verified that if r = r " then

(4.6) wm (re '~ tan-l (r-~) < 4 r ~ 71- r m ~

Consider the ha rmon ic funct ion g,, in ~ " given by

g,~(re 'e) ems inOAr~L(rm)+ (r") L(rm) trm) Ltr,. = K '~ + K ' .... ~-" )w"(re '~ sin 7rA


By (4.5) it is clear that f Gm I =< gm on 0a//m and so t G,, gm in ~ But by (4.6) we

have for r~I , , (~)=[~r- 'r , , ,o ' rm], o->1 ,

g,~(re'~ ' {e , r*L(r , , )+(~)"r~L(r ,~ )+(r -~ , ) l - " r"L(rm)}

(4.7) <__ e "r"L(r,~)

where K ' is a constant independent of m and 0 and, by (2.11), e ' - - ~ 0 as m ~ .

Since I G~ [=< g,~, (4.7), (2.11), (2.12) and a diagonalization prove the lemma.

Since v is subharmonic with boundary values at most those of H, Lemma 4.1

implies

(4.8) v ( r e ' ~ (r--->%r~I,~,O<=O<--_~)

where the o(1)-->0 uniformly in 0.

5. Petrenko's formula and the proof Of (2.15)

In order to use the upper bound on v given by (4.8) we will introduce some local

approximations to v(re '~ which are obtained by integrating log I f[ over a fixed

interval of length 20. In fact we will asymptotically evaluate these approximations.

To do that as well as to obtain (2.15), we state and use a variant of Petrenko's

formula established by Edrei and Fuchs [6, lemma 11.2].

L e m m a 5.1. Let f be a transcendental entire function with f(O) = 1. Let {R "}

and {R~} be sequences such that R'--->~ as m- ->% 4 R " < R~ , let {o)m} be a

sequence of arguments, and suppose that 0 < 3' <-- 1. Introduce the sectors

Sm={z =re '~ :R'<=r<=R",lO-6oml<-_ yTr} (5.1)

and set

(5.2)

and

(5.3)

Then

(5.4)

t l/Vr~/V X(t , r) = (t,/~ + r,/~)2

izl. +lcl ',, H(z , C) = log [z ''' - C TM [

l o g l f ( r e ' ~ ) l + ~" H(a,e- '~-,r) a j E S m

n';.

= " 7 Y n;.,

ro m + y r r

loglf(te'~ S(t,r)T E(R',r,R') o~m - - 3,~r


where {a~} denotes the zeros of f and where

'} I E ( R s logM(2R ',)+ IogM(2R"

(5.5) (A > 0 an absolute constant)

provided that 2R " <= r ~ �89 ".

In (5.2)-(5.4) set y = fl/,r, where, as usual, 0</3A < ~r. Set

(5.6) r = rm, R " = r ' , R " = �89

(cf. (2.11)) and let o)m be any sequence satisfying

(5.7) {f(r..e '~-)l = M(r..).

We shall keep this sequence fixed throughout the remainder of the proof of Theorem 2.

Observe that the definition of v( te*) and the choice of 2/ imply that

1 f ioglf(te,e)[dO 1 f log l f ( t#~ dO 2= j = ~ j

(5.8) <- ~ v(te*). --2"n"

Also observe that (2.4), (2.11), (4.2), and the fact that 1/~ = rr//3 > A imply that

(5.9) [E(r',., r.,~r".)t = o(logM(rm)) (m --->oo).

Thus (5.4)-(5.9) imply that

(5.10) logM(rm)+ ~'~ H(a,e "~-,rm)--<2-~T2 f " X dr+ ~,~s. -" v(te ) (t, rm)T o(logM(rm)). G,

Now, (2.4), (4.8), (5.2) and an easy contour integration show that

1 v(te,~)X(t, rm ) 2zry 2 r;.

12 [sin~x + (r.) f rx(t, r.) dt "~ t 7rA •..] L t

0

(5.11) = (1 + n,)r~L(rm)

= (1 + ~%)log M(;~),

where rim-"~0 as m-->~o. Thus (5.10) and (5.11) give

(5.12) logM(r.~)+ ~ H(aje -~,., rm)= < (1 + o(1))logM(r,,) aI(ES m

( m -----> oo),

ASYMPTOTIC B E H A V I O R OF FUNCTIONS 141

Since H(z, r) >- 0, (5.12) implies

(5.13) H(aje -'~-, r,~) = o(log M(r,,)) (/1"I ~ o0).

If, for t r > l and 0 < a < /3 ( = 7r~/), we set (as in (2.13)) ~m(cr) = ~r~ (O'; to,~, Or), then some easy estimates (cf. [6, lemma 12.1, (12.6)]) show that there is a positive constant .,~ = fi~ (/3 - a, o-) such that

H(aje , rm) = A n ( ~ , ( ~ ) ) . (5.14) , , ~ ( ~ ) -,o. >

NOW (5.1), (5.6), and (2.11) imply that N~(~r) C_ Sm for m => m~ = m~(tr). Thus (2.15) follows from (5.13) and (5.14).

6. L oc a l a p p r o x i m a t i o n s to the f u n c t i o n v (z)

To complete the proof of Theorem 2 we still need to prove (2.16). The proof commences here and will be completed in w

For 0 =< 0 _--< ~, consider the functions ~m+O

/*

(6.1) v , tre'~ = J logff(re'*)ld6, m = l , 2 , . . - ~ m - 0

where {to,,} is the sequence of arguments defined by (5.7). By definition v,~ (re '~ <-_ v(re'~ therefore, if r E Ira, and 0---O-</3, (4.8) implies that

(6.2) v,~(re~~ ) (r/,~ ---~ 0 as m ---~ ~).

In fact, if or > 1,

(6.3) o , ( r e ' ~ (r--~oo, rEIm(tr),O<=O<<-/3)

where o(1)---,0 uniformly in 0. Once this equality is established it will follow that v,,(re '~ is asymptotic to v(re 'e) for r CIm(o'), 0<= 0 <=/3, and is therefore the desired local approximation to v(re'e).

If (6.3) were false, there would exist tr > 1, ~ > 0, an unbounded sequence M of positive integers, and sequences {t~,}, {or,} with t,. E/ , , (or) , 0 < ~,~ =</3, such that for r n ~ M ,

v , ( t , e . ) < [ 2 , ] t~L(rm). (6.4) ,~ sin area = A

One can prove as in [6, lemma 6.1] that if (r > 1 and cr -1 =< tilt2 <= cr then

Ivm (t,e'~ _ v,~ (t2e ~~ <= K log M(R, f) 1

x { ( o ' - 1) (1 + l o g + ~ _ 1) + 102- 01' (1 + log+ , 0 _ 01j) }

142 J. R O S S I A N D J. W I L L I A M S O N

where tj -< R/4 , j = 1, 2 and K is a positive constant. By using this, (4.2), and (6.4),

we can easily adapt lemma 6.2 of [6] to the functions vm to show that there exist o-1, 1 < o-, and ~b, 0 < ~b-</3, such that

(6.5) v,(te'~)<= [2sinA~bA-~3] PL(rm)

provided that tm/O'l <= t <= o'ltm, m CAt, m >- too. We will now use Petrenko 's formula to show that (6.5) leads to a contradiction.

If we set T = th/Tr in (5.2) and (5.4) and choose r = r,,, R 'm, R " , and to,, as in (5.6)

and (5.7), then in view of (5.9) (which still holds since 1/3, = 7r/~b => 7r//3 > A) and

(5.13), (5.4) becomes

(6.6) log M(r,~) = 2---~y2 f v,,(te'*)X(t,r,)-~+o(logM(r,,)). ,,;,

If we denote by J(rm) the integral on the right-hand side of (6.6), then by (6.2) and

(6.5), m E ~ , m _-> mo, implies

12[s in thh+ ] J ( r ' ) = < - ~ [ 7rA r/,, L(r,~).Jl(r,.)-[~l+3,2rl.,lL(r.)J2(rm) (6.7)

where

and

�89

f d, J2(r.. ) = tAX(t, r,.) -i-' tm/o" 1

~1 = ~/6zr3' 2.

Since Y = ~b/Tr, (5.2) implies

(6.8) Jl(r,,)<= f PX(t, rm) dt_ 3'27rA t - sin ~bh r m.

0

Also,

tr l t m / r m

f s~X(s, 1) d s - ~ (6.9) J2(rm) = rm s - rmxm. t m / { o ' l r m )

Since oh > 1 and tm E/ , , (o r ) = [o'- 'r, , , trrm],


(6.10) X-, --> 6 > 0 for m E ~ .

Thus, if m E ~ , m ~ m0, (6.6)--(6.10) imply that

(6.11) (1 + o(1) ) logM(rm) <- (1 - ~)r~L(r , . )

where ~ = �89 > 0. In view of (2.4), this is not possible. Consequently, (6.5) is false

and (6.3) true.

7. An asymmetr i c vers ion of (2.16)

In this section we consider important harmonic approximations to the functions

v,~(re '~ for 0 = < 0 =</3 and suitable r.

We begin by using the "scarcity" of zeros of f in the sectors 9,, (or; win, a ) ((2.15))

to define functions qJm which are harmonic in these sectors and which, except on

"small" exceptional sets, approximate log Ill . For 0 < a </3, an obvious diagonal-

ization of (2.15) and, if necessary, a change in notation, yields

(7.1) n., = n ( 9 , . ( t o , . , a ) , O ) = o ( l o g M ( r , . ) ) (m--~o~).

Now, let {p,.} be any sequence of positive numbers satisfying

(7.2) p,. ----> 0%

(7.3) r " < _ r,.e -p", < r,.e "m =< r" ,

(7.4) n,.p,. = o (log M(r. , ) ) (m ---> ~).

The relation (7.4) is possible in view of (7.1). Redefine r ' , r " as r ' = e-Omr,.,

r " = e",r,, and, with this new notation set, as before, I,. = [ r ' , r"], A = U~.=~ Ira,

and

(7.5)

Now define

9 , . = 9 , . ( o J , . , a ) = { r e '~ :rEIm, lO-~ml<c~}.

(7.6) P . . ( z )= 1-I ( 1 - z / a j )

where the aj are zeros of [, and

(7.7) ~b,. (z) = log I f ( z ) [ - log[P . (z)]

Clearly the functions q,m are harmonic in the sectors 9., and are subharmonic in the

plane (note that tbm(0)= 0, since f (0 )= 1). The following elementary lemma

describes two important properties of the qJ,..

L e m m a 7.1. I f ~r > 1 and M(r , qJ,,) = maxizr=,~,.(z), then

(a) ~,, (z) = log [ f ( z ) l + o (log M ( r ) )


as r --~ o% r -- I z [ E / ~ (tr), z ~ ~m where ~m is a union of finitely many disks, the sum of whose radii is o(r~,) as m-~oo, and

(b) M(r ,d /m)=( l+o(1) ) logM(r ) (r-~ % r ~ / ~ (tr)).

We will omit the details of the proof of Lemma 7.1 but remark that part (a) can

be obtained from lemma 1.1 of [7], (2.4), and (4.2). Part (b) follows from the fact

that M(r, ~ ) is an increasing function of r, the nature of the exceptional set ~,~, and

a routine argument. Our next few results are about harmonic approximations to the functions vm. For

0 _-< 0 _-< 7r, define

~ m + 0

(7.8) u"(re '~ I qJ"(re'~)d~~ a~ m - - 0

Note that um (0) -- 0 since ~b,~ (0) -- 0 and that for notational simplicity we can write

(7.9) u,. (re '~ = i ~" (re'~)dq~ - - 0

where

(7.10) ~bm (re '~ ) = d/,. (re"~§

Observe that ~,. is subharmonic in the plane and harmonic in ~,. where

~,. = ~ . , ( O , a ) = {re '~ : r E I , . , l O l < o ~ </3}.

Our next lemma describes some important properties of u,. and ~,..

L e m m a 7.2. (a) The functions u,. are harmonic in ~ +..= {re '~ : r E L . O < 0 <

~ </3} . (b) The functions (O,. possess the following asymmetric property: I f tr > 2,

r E I , . i 0 < 2 7 7 = 0 _ < (~o'), and < /3-271, then

~r.(re'~ ~.:(re -'~) = [2cos 0A + o(1)]logM(r)

where o(1)---~0 uniformly in 0 as r---~% r G I,.(�89

P r o o f . (a) The proof of (a) follows directly from the harmonicity of ~,..

(b) First note that by (7.4), (7.6), and a simple application of Jensen's theorem

(of. [7, lemma 1.1]),

lr

(7.11) 1 f 2-~ [l~176 IdO =o(logM(rm)) (r----~oo, r~I ,~) . - - o r

ASYMPTOTIC BEHAVIOR OF FUNCFIONS 145

Thus (7.7), (7.11), and (6.1) give

+ 0

u"(re'~ i ~b"(re'~')dcb o~ m - o

oa + o r a

= I i~ +o(logM(r,.)) (7.12)

w m - O

= v,,(re'~ (r--->~,r El,.).

Therefore, if o- > 2 and if re'~ ~ ~,, (o'; o~,,,/3), (4.2) and (6.3) imply that

(7.13) u,, (re'~ = [ 2 s~ O~t + o(X ) ] r~ L (r,, ).

Since o(1)--->0 uniformly in 0, 0 - < 0 <=/3, as r-->~, r E I,, (o-) and since u,(re '~ and r ~ sin 0)t are harmonic in ~+,,, it follows from (7.13) that for r E I,, (o-/2) and

0<27/_-__ 0_<-/3 -2r t ,

(7.14) 0u,, (re'6) = [2 cos 0)t + o (1)]rAL(r,,,) a0

where o(1)--->0 uniformly in 0.

Equations (4.2), (7.9), and (7.14) then imply that

~,.(re'~ ~,.(re -'~ = [ 2 c o s 0X + o(1)]logM(r)

for r E I,, (o-/2) and 0 < 2~q _----- 0 =</3 - 2r/. This proves (b).

8 . C o n d i t i o n a l p r o o f o f ( 2 . 1 6 )

.At this point we note that if we knew that the ~,, were even functions of 0 (2.16)

would follow almost immediately from (7.10), part (a) of Lemma 7.1 and part (b) of Lemma 7.2. In order to establish the symmetry of the ~ , we will adapt methods of

Baernstein [I].

Let ~ > 0 be as in Lemma 7.2 and set

(8.1) /3, = 13 - 2ft.

As in [1] define

(8.2) w(re '~ = sup f loglf(re~')tdO (0< r,0=< 0 _-</3,/2) E

where the sup is taken over all sets E of the following form:

(8.3) E = [a,, bl] U [a2, b~] U [a3, b3]

146

with

J. R O S S I A N D J. W I L L I A M S O N

al _--< bl --< a2 _-< b2 -< a3 -< b3,

b 2 - a 2 = 2 0 , ( b l - a , ) + ( b 3 - a 3 ) = 2 0 ,

a2-- bl = a 3 - b2 = /31 - 2 0 .

It is shown in [1, p. 193] that w is subharmonic when 0 < 0 </3~/2 and cont inuous

when 0 - < 0-</31/2. We define

Hi(re '~ = H (r exp{ i (/3d2 + 0)}) - H (r exp{ i (/3d2 - 0)})

for 0 _-< 0 _-</31/2 where H is as in w Recall the definitions (3.5) and (3.8) of v and H

respectively. We have v(re '~) = H(re'O), v ( r )= H ( r ) = 0, and since v is subhar-

monic and H harmonic we deduce v (re '~,) <~ H(re'O,). Thus

(8.4) w (re'~'/2) = v (re'~') -< H(re'~') = Hl(re'~'/2).

Fur the rmore wl(r) = Hi(r) = 0. A n d so since/-/1 is ha rmonic in 0 < 0 </31/2 and w

is subharmonic there, we have

(8.5) w (re ,o) < H,(re '~) (0 <= 0 <=/3112).

Recall the definitions (6.1) and (7.9) of vm and u,,. By L e m m a 4.1, the remark

following (6.3) and (7.13), we have that locally H - v - v,, - u,,. This, and the fact

that w(reiO,/2) = v(re '~,) and Hi(re '~r = H(re'O), make it clear that there exists

r/,, ~ 0 as m ~ ~ such that for r E I,, (or)

w(re'~,/2) >_ H~(re'~,) + n,.r ~L (r,, ) (8.6)

and

,81

(8.7) w(re'~r <- f 6(re'*)dqb + 71,,r~L(r,,). -lSn

Fur the rmore by (4.2), L e m m a 7.1(b), (7.11) and the definit ion (7.10) of ~,~ we can

find r/'----~0 as m---~oo such that for r E I , , ( t r )

t A M ( r , ~ m ) = (1 + rl~)r L ( r m ) (8.8)

and

E-~ m E

where E is as in the definition of w and E - to,. = {0 : 0 - to,, E E}.

Also by the definition of H1 and L e m m a 4.1, we may find r/"---~ 0 as m ---~ oo such

that for r E L, (or) and 0 =< 0 _-</31/2

(8.1o)

where

(8.11)

A S Y M P T O T I C B E H A V I O R OF F U N C T I O N S

f H,(re '~ h.. (re'~ < " ~ = n,~r L(r.,)

h,. (re ,o) = 23. ~'[sin .k (/3J2 + O) - sin h (/3~t2 - O)]r*L(r,.).

147

where

~ . , (r) = sup r176

Dividing both sides of the above inequali ty b y / 3 ~ - 20., and taking into account

(8.12) and (8.13) we obta in for r E 1,,(~r)

Let e,, = max(r/, , , 7'-,, r l " ) and let 0,, > 0 be chosen such that

(8.12) O= 2'/3,t2

and

(8.13) e,, 1(/3I/2 - & ) - - * 0

a s m - - + m .

As in [11 fix E = E(O) = [ - /31 , -/31] tO [ - 20,0] U [/31-20,/3,1. No te that E has

the form (8.3) and thus so does E - to,,. By definition (8.2) of w we have

(8.14) w(re'~ loglf(re *)ld4 .

So by (8.4)-(8.11) and (8 .14 )we obtain for r E I.,(ar)

2h -~{sin/3~A - [sin(/3 J2 + 0.,)~ - sin(/3d2 - 0.~)~ ]}r~L (r,.)

= (2~ -~ sin ~81~ - h,. (re'~

<= Hi(re ,aj2) _ Hi(re to,. ) + 2e,.r~L (r'-)

<= w (re"J2) _ w (re '~ + 38"-rAL (r,.)

t3~

f ~"(re'~) ddp- f ~"(re'~) d4a+4c"r;L(r") - & ~(om)

= f ~"( re'*)dcb+ f (O"(re")dq~+4e'rXL(r') -,a~ 0

(/3, - 20~) ( ~ (r) + M(r , ~ ) ) + 4emr~k(r,.)

(~, - 20,.)(~'- (r) + r~L (r,.)) + 5e,,,r% (rm)


(8.15) (cos/3,A + l)r*L(rm)<= ~,.(r)+ r~L(rm)+ e'r*L(r,,)

where e'----~0 as m---~ . Thus (8.15) implies

(8.16) ~, , (r) => (cos/3,A -e')r~L(r,.) (r~I,,(cr)).

We now need the following

L e m m a 8.1. I[ o'> 1 then ~(re-"')>=(cos/3,A +o(I))r~L(rm) where r---~, r E I,.(oO.

Assuming that the lemma is true we must also have that

~,, (re '~,) >= (cos/3,A + o(1))r%(r,,)

by repeating the same arguments with E(O) = [ -/31, - /3 , + 20] U [0, 20] U [/31,/3,].

By Lemma 7.2(b) this means

(8.17) d,,(re§ = (cos/3,A +o(I))r*i (r,,) (r-->o%r~I,,(o-)).

Also by Lemma 7.1(b), Lemma 7.2(b) and (4.2) we have

M(r,(b~)=(l+o(1))r~L(rm) (r-->o%rEIm(tr)) (8.18)

and

(8.19) ~,,, (r) = (1 + o (1))rAL (r,,) (r ~ ~, r • I,. (or)).

Now we "diagonalize" (8.17)-(8.19) and replace I,. (or) with intervals, say I,, of

the form (2.11) and (2.12). Since ~,. is harmonic when r ~ I,., -/31 < 0 < 131,

continuous for -/31 ~ 0 =</31, with/3, < rr/A and (8.17)-(8.19) hold, we obtain in

the same way as in the proof of Lemma 4.1 that for or > 1

(8.20) ~,,(re'~ ~-o(1))r*L(r,.) (r---~o%rEh,(o'),O<=O<=/30

where o(1)--*0 uniformly in 0. Recalling the definition (7.10) of ~., and Lemma 7.1(a) we have that, except for

the exceptional set of the lemma,

log If(re'~ ~ (cos(0 - to,)A + o(1))r*L(r,)

holds for r ~ Im (or), 0 = < 0 =</31 where o(1)---~ 0 uniformly in 0. Recalling (4.2) and

(8.1) we obtain (2.16).

9. C o m p l e t i o n of the P r o o f of T h e o r e m 2

To complete the proof of Theorem 2 we only need to prove Lemma 8.1. To do

this we use the following version of a lemma of Fuchs [9] found in [17].

L e m m a 9.1. Let g(z) be an entire [unction of finite order A. Given ~ > 0 ,


cr >- 1 + ~, 8(0 < 8 < ~) and a sequence {r,.} of P61ya peaks of order )t for log M (r, g),

there is a constant C = C(A, if, or) such that for all r ~ U~=l I,, (~r) - F, dens F < if,

and every interval J of length 8

Recall that

f I g ~ d S < C ( 8 1 o g + l ) logM(r ,g) . r g(re,O) = J

- - meas F N [1, r] dens F = lim

r-~ r - 1

Also recall that {rm} is a sequence of P61ya peaks for log M(r, g) provided that there

exist sequences {r'} and {r"} such that as m - - - ~

t O0 ! O~ r m / r r a - - 9 , 0(2, r ,,---~ , r,, / r .~---~ , " '

and

l ogM(r ,g )<=( l+o(1 ) ) ( r / r , , f l ogM(r , , g ) (r'<=r,~<=r").

Since (2.4) and (4.2) hold, the sequence {rm} (in G ) that we have been considering is

a sequence of P61ya peaks for log M ( r ) = log M(r, f ) .

Now let

i ( 0 + ~ ) ~,.(re~O)= f(re - )

Pm (re,~ ))

where the o~,~ are as in (5.7) and the Pm as in (7.6). Then f,, is an entire function,

m = 1 ,2 , - - - , and since (7.7) and (7.10) imply

t~, (re '~ = log l fro (re'~

it follows from (7.10) and Lemma 7.1(b) that {rm} is a sequence of P61ya peaks for logM(r, fm)=M(r ,~m), r e = l , 2 , . . . . Thus we can apply Lemma 9.1 to fro,

m = 1 , 2 , - - . .

R e m a r k 9.2. The exceptional set F,~ for f,~ in Lemma 9.1 arises from

Cartan's Lemma. Since by definition of Pro, fm has no zeros in Vm = {re 'e : r E L , I O I < - / 3 ~ - - /3 -2"0}, F m n vm = 0 . Thus i f F i sanarcin VmandJ is the interval of 0 with re'~ E F, the conclusion of Lemma 9.1 holds with no exceptional set.

Now assume that Lemma 8.1 is false. Then there exist cr > 1, ~ > 0 and a

sequence s,. where m E d~, an unbounded sequence of positive integers, such that s,~ ~[o'-lr,,,crr,,.] and

(9.1) ~,~ (s,~e-~S~) <= (cos/31A - ~)s ~L (rm).


For m E ~,/, m => mo we see f rom (8.16) and (9.1) that there exists an interval

Jr. C__ [ --/3,, --20,.] with endpoints a.~ and yr" satisfying

{ 6.,(sr"e '% ) = (cos/3,A - ~/2)s~L(r,.),

(9.2) ~0., (s,.e ' ,- ) = (cos t31X - sC)s ~L (r,.).

Let Fr" be the circular arc connect ing sr"e '~ to s,.e ',~ such that the argument of

each point of Fr" is contained in Jr". Then by (9.2) we have

A -~ sr"L(rr")= ~.~(s,.e"-) - ~., (s~,e"-)

(9.3) = Re f / ' ( z ) d z f.(z)

Fm

ff , ,of {r"(sme ) dO. <=

Jr.

Since 20r"/~fl, and Jm C [ - f l , , - 2 0 , , ] we see that the length of Jr" = 8m-->0 as

m--> oo. Fu r the rmore Fr" C Vr", where Vm is as in R e m a r k 9.2. Thus by L e m m a 9.1

and R e m a r k 9.2, the last integral in (9.3) is o(logM(sm,fr"))=o(M(sm,~r")) as

m --~ oo. Then by (9.3), L e m m a 7.1(b), (4.2) and the definit ion (7.10) of r we have

g s~L(rr") = o(s~L(rr")) 2

which is not possible since ~: > 0. This contradict ion implies that Lemma 8.1 is

indeed true. Thus, as ment ioned at the beginning of this section, the proof of

T h e o r e m 2 is now complete .

I 0 . P r o o f o f T h e o r e m 3

Our p roof is based on a combina t ion of a " loca l ized" version of a classical

t auber ian- theorem of Valiron [19, t heo rem A; 18, corol lary 1.1] and a theorem of

Whee le r [19, t heorem 1]. We state it as

T h e o r e m C. Suppose that g is a canonical product of genus q and order 3.,

q < A < q + 1, with only negative zeros. Suppose that there exists a se tA oftheform

(10.1) A = I,.J [cr",dr~], m = l

where cr", dm/cm-+ oo as m - * ~ , and a sequence {co,,} such that

(10.2) log[g(re '~- ) l - logM(r ,g ) (r--~,rE[cr",dr"]) .

desired, then

(io.3)

implies

A S Y M P T O T I C B E H A V I O R OF FUNCTIONS 151

I f L(r) grows slowly on A and if A~ = [(r,,cm, (r-~dm], where om-- )~ as slowly as

l o g M ( r , g ) - r ~ L ( r ) ( r - - * ~ , r @ A )

(10.4) n(r) = n(r,O; g ) - Isin ~r)tl r~L(r) 7/"

Now denote by {t.~} any sequence such that

cr,.c,. -< t.~ =< 6r~,tdm and tm d,. o - , . c , . ' o -rot , .

Also, for ~r > 1, set

(10.5) J~(cr)=[o'-~tm,o'tm] and A(~r)=

(r--->~,r ~ A,) .

- - - - ) ~ a s m - - , ~ .

U jm (~). m = l

l o g M ( r , f ) = ( l + o ( 1 ) ) l o g M ( r , g ) (r-->~).

Now let {r,.} be a sequence in G satisfying (2.9), let o- > 1, and let the sealuence {oJ,, }

satisfy

(10.10) [f(r.,e '~'- )l = M(r,., f) .

In view of (10.9) and the fact that g ( ~ . ) = g ( z ) we can assume to,, > - 6 ,

0 < 8 < 7r/2, without loss of generality. If we further assume that

(10.s)

so that

(10.9)

If we use the well-known Valiron integral representation of log]g(re'~ [ 0 l < 7r,

together with Theorem C, then standard arguments readily yield

C o r o l l a r y D. Let the assumptions of Theorem C be unchamged. Then, ]:or

o- > 1, (10.2) and (10.3) imply

(10.6) loglg(re'~ ( - 1)q[cos 0A +o(1)]r~L(r) (r ~ A(o'))

uniformly in r and O, ] 0 ] <= 7r - 8 < 7r, as r ---) ~ in A(o-).

Suppose now that f satisfies the hypotheses of Theorem 3 and f(0) = 1. Then we

can write

(10.7) f ( z ) = e P(~)g(z)

where P ( z ) is a polynomial of degree less than A and g( z ) is a canonical product of

genus q = [A ]. As is well known,

r q = o(logM(r, g)) (r---)~)


lim m,. < 7r, (10.11) , , ~

then (2.16) and the fact that the exceptional disks are centered at points on the

negative axis implies

(10.12) log If(re'~m)l = (1 + o(1))logM(r,f) (r --~oo, r ~ I,, Or)).

In view of (4.2) and (10.9), (10.12) implies

log I g(re '~-)1 = (1 + o (1))log M(r, g)

(10.13) =(l+o(1))r*L(r , , ) (r - - -~ ,rEI , , (~r)) .

Since cr is arbitrary subject to o- > 1, a diagonalization of (10.13) yields sequences

c,,, d,, where c,,, d,./c,,--~oo such that if A = U~,=I [c,.,d,,],

log I g(re '~- )l = (1 + o (1))log M(r, g) (10.14)

=( l +o(1))r*L(rm) (r~oo, rE[c, . ,d, ,]) .

Theorem 3 then follows from (10.7), (10.14) and Corollary D.

Thus, the proof of Theorem 3 will be complete once we establish (10.11). In fact,

we will show that

(10.15) lim tom _-< 7r - ft. r n ~

If (10.15) were false, there would exist e > 0 and an unbounded set of positive

integers d~ such that

(10.16) ~ r - f l + e _-__ to,. _-__ Ir (m Ed~).

If tr > 1 and A~(tr) -- UmE~ [o'-lrm, O'rr, ], then (2.15), (10.16), and the fact that [ has

only negative zeros imply

(10.17) n(r,O;f) = n(r) = o(logM(rm)) = o(rXL(r,.)) ( r - . o% r E A~(~r)).

Diagonalizing (10.17) in a suitable way, we can find sequences {r'} and {r~}

satisfying (2.11) and {era}, em-~0 as m--.oo such that if A~ = U m e ~ [ r ' , r~,],

n(r)<=emr~L(rm) (r--) o% r ~ A1). (10.18)

But this means

(10.19) N(r) = o(r*L(rm)) (r.---~oo, r E A2)

where A2 has the same structure as A1,AzC_A1. To see this let r ~ [o--lr,,, o-r,,],

o- > 1. Then by (10.18) and (4.2) we have


N ( r ) = N ( r ' ) + I dt t

r;.

_= l o g M ( r ' ) + e,.X ' r~L(r~)

<-- r" L(rm) + emA -' rXL(rm)

= o(r~L(rm)).

Since this holds for eve ry or > 1, we r e -d i agona l i ze to ob ta in A2.

Us ing a r e p r e s e n t a t i o n for the Nevan l inna charac te r i s t i c T ( r ) = T(r, [ ) in t e rms

of N ( r ) ([12, l e m m a 1]), it is easy to show tha t (10.19) impl ies

(10.20) T(r ) = 0 (r~L (r,,)) (r ~ o0, r E A2(tr)).

Since log M ( r ) <= 3 T ( 2 r ) for r > 0, (10.20) is i n c o m p a t i b l e with (4.2). Thus , (10.16) is

false and (10.15) t rue , and the p roof of T h e o r e m 3 is comple te .

ACKNOWLEDGMENTS

W e would l ike to t h a n k AI Bae rns t e in for severa l he lpfu l c o m m e n t s on the

resul ts in the first a u t h o r ' s d i sse r ta t ion on which this p a p e r is based , and we wou ld

espec ia l ly like to t h a n k D a n Shea for b r ing ing Ba e rns t e in ' s ques t ion to our

a t t en t i on while visi t ing Hawa i i in the Spr ing of 1979 and for m a n y va luab le

sugges t ions dur ing the course of our research . F ina l ly we thank the r e fe ree , W a l t e r

H a y m a n , for sugges t ing r eo rgan iza t i on of the or ig ina l manusc r ip t which we hope

has m a d e this r a the r l eng thy p a p e r eas ier to read .

REFERENCES

1. A. Baernstein II, A generalization of the cos ~'p theorem, Trans. Amer. Math. Soc. 193 (t974), 181-197.

2. R. P. Boas, Jr., Entire Functions, Academic Press, New York, 1954. 3. D. Drasin, A flexible proximate order, Bull. London Math. Soc. 6 (1974), 129-135. 4. D. Drasin and D. Shea, Convolution inequalities, regular variation and exceptional sets, J. Analyse

Math. 29 (1976), 232-293. 5. A. Edrei, A local[orm o[the Phragmen-Lindel6[indicator, Mathematika 17 (1970), 149-172. 6. A. Edrei and W. H..;. Fuchs, Asymptotic behaviour of meromorphic [unctions with extremal

spread I, Ann. Acad. Sci. Fenn. Ser. A1 2 (1976), 67-111. 7. A. Edrei and W. H. J. Fuchs, Asymptotic behaviour of meromorphic functions with extremal

spread ll, Ann. Acad, Sci, Fenn. Ser. A1 3 (1977), 141-168. 8. M. Ess6n, The cos 7rA theorem, Lecture Notes in Mathematics 467, Springer-Verlag, Berlin, 1975. 9. W. H. J. Fuchs, Proof of a conjecture of G. P61ya concerning gap series, Illinois J. Math. 7 (1963),

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A. J. Macintyre, Athens, Ohio, 1970, pp. 149-170. 11. W.K. Hayman and P. B. Kennedy, Subharmonic Functions, Academic Press, London, 1976. 12. S. Hellerstein, D. Shea and J. Williamson, A Tauberian theorem characterising the regularity of

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13. B. Kjellberg, On the minimum modulus of entire functtons of lower order less than one, Math Scand. 8 (1960), 189-197.

14. B. Kjellberg, A theorem on the minimum modulus of entire functions, Math Scand. 12 (1963), 5-11,

15. B. Ja. Levin, Distribution of zeros of entire functions, GITTL, Moscow, 1956; English Transl.: Trans. Math. Monographs, Vol. 5, Amer. Math. Soc., Providence, R,I., 1964.

16. J. Rossi, Two problems in entire function theory : Reciprocals of entire functions of infinite order and functions extrcmal [or the cos/3A theorem, Ph.D. thesis, University of Hawaii, 1980.

17. J. Rossi and A Weitsman, A unified approach to certain questions in value distribution theory, J. London Math. Soc., to appear.

18. R. Wheeler, Some extensions of Valiron ' s Tauberian theorem for entire functions, Ph.D. thesis, University of Wisconsin, 1971.

19. R. Wheeler, A maximum modulus analogue of Vahron ' s Tauberian theorem for entire functions, Quart. J. Math. Oxford Ser. (2) 24 (1973), 315-331.

DEPARTMENT OF MATHEMATICS PURDUE UNIVERSITY

W. LAFAYETTE, IN 47907 USA

DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAWAII

HONOLULU, HI 96822 USA

Current address of first author DEPARTMENT OF MATHEMATICS

VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY BLACKSBURG, VA 24061 USA

(Received July 1, 1982 and in revised form July 10, 1983)

Documents

The asymptotic behavior of functions extremal for Baernstein’s cos Βλ theorem