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Complex Variables
ISSN: 0278-1077 (Print) 1563-5066 (Online) Journal homepage: https://www.tandfonline.com/loi/gcov19
Distance between a Maximum Modulus Point and
Zero Set of an Entire Function
Iossif Ostrovskii & Adem Ersin Üreyen
To cite this article: Iossif Ostrovskii & Adem Ersin Üreyen (2003) Distance between a Maximum
Modulus Point and Zero Set of an Entire Function, Complex Variables, 48:7, 583-598, DOI: 10.1080/0278107031000120431
To link to this article: https://doi.org/10.1080/0278107031000120431
Published online: 15 Sep 2010.
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Distance between a Maximum Modulus Point
and Zero Set of an Entire Function
IOSSIF OSTROVSKIIa,b,* and ADEM ERSIN REYENa
aDepartment of Mathematics, Bilkent University, 06800 Bilkent, Ankara,Turkey; bVerkin Institute of LowTemperature Physics and Engineering, 61103 Kharkov, Ukraine
Communicated by M. Essen
(Received 6 December 2002; In final form 15 March 2003)
Let f be an entire function of finite positive order. A maximum modulus point is a point w such that jf ðwÞj ¼maxfj f ðzÞj: jzj ¼ jwjg. We obtain lower bounds for the distance between a maximum modulus point w and the zero set of f . These bounds are valid for all sufficiently large values of jwj.
Keywords:Entire function; Order; Proximate order; Strong proximate order; Type
2000 AMS Subject Classification:30D20
1. INTRODUCTION
Let f be an entire function. A point w 2 C is called a maximum modulus point if
jf ðwÞj ¼ Mðjwj, f Þ,
where
Mðr, f Þ ¼ max
jzj¼rjf ðzÞj:
We denote by Rðw, f Þ the distance between a maximum modulus point w and the zero set of f , i.e.
Rðw, f Þ ¼ inffjw zj: f ðzÞ ¼ 0g:
*Corresponding author. E-mail: iossif@fen.bilkent.edu.tr; OSTROVSKII@ilt.kharov.ua
ISSN 0278-1077 print: ISSN 1563-5066 onlineß 2003 Taylor & Francis Ltd DOI: 10.1080/0278107031000120431
Following Macintyre [3], let us introduce the nondecreasing function Kðr, f Þ ¼ d
dlog rlog Mðr, f Þ, r >0, (we take the right derivative at fracture points of log Mðr, f ÞÞ.
Macintyre [3] showed that the well-known Wiman–Valiron formula (see, e.g. [1], p.22) describing the behavior of f in a neighborhood of a maximum modulus point remains valid with Kðr, f Þ instead of the central index of f . Namely, Macintyre showed that
f ðzÞ ¼ f ðwÞ z w Kðjwj, f Þ
ð1 þ oð1ÞÞ, w ! 1, jwj =2E, ð1Þ where E ð0, 1Þ is an exceptional set and z lies in a (depending on w) neighbor-hood of w.
In Macintyre’s proof of formula (1) estimates of Rðw, f Þ from below played an important role. We formulate these estimates as the following separate theorem. THEOREMA ([3])
(i) The following inequality holds
lim sup
jwj!1
1
jwjRðw, f Þðlog Mðjwj, f ÞÞ
1=2>0: ð2Þ
(ii) For each >0 the following inequality holds
lim inf jwj!1 jwj =2A 1 jwjRðw, f Þðlog Mðjwj, f ÞÞ ð1=2Þþ>0, ð3Þ
where A Rþ is such that
Z
A
dt
t < 1: ð4Þ
The inequality (2) gives an asymptotic bound for Rðw, f Þ from below only on a sequence of values of jwj ! 1. The inequality (3) gives a bit less precise bound which is valid outside of a ‘‘small’’ set. In this article we are going to show that bounds for Rðw, f Þ given by (2) and (3) cannot be valid in general without exceptional sets at all and find in some sense unimprovable bounds valid for all sufficiently large values of jwj.
Note that far reaching generalizations of (1)–(3) to functions analytic in a half-plane, multi-valued functions and entire functions of several variables were obtained by Sh. Strelitz [6]. We think that similar generalizations are possible for our results as well, but we do not touch them here and restrict ourselves to entire functions of one variable and finite positive order.
Note also that, for some specific parametrical families f fg, 0 < < 1, of entire
for Rðw, fÞfrom below play an important role in the theory of phase transitions (see [5],
Chap. 3). Nevertheless, we cannot extract from our results any useful consequences for this theory because our approach does not permit to take into account asymptotical behavior in parameter ! 1.
2. STATEMENT OF RESULTS
Let f be an entire function of order . We assume that 0 < < 1.
To state our results we need the notions of proximate order ([4], p.31) and strong proximate order ([4], p.41).
Remind that a proximate order is a function ðrÞ 2 C1ðR
þÞsuch that
(i) 9limr!1ðrÞ ¼ ð>0Þ,
(ii) limr!10ðrÞrlog r ¼ 0.
Note that ([4], p.32)
lim
r!1r
ðrÞðkrÞðkrÞ¼
k, ð5Þ
uniformly on each interval 0 < a k b < 1.
By Valiron’s theorem ([4], p.35) any entire function f of order has its own proximate order, that is there exists such a proximate order ðrÞ ! that the type
:¼ lim sup
r!1
rðrÞlog Mðr, f Þ
is finite and positive. We denote by ½ðrÞ, the set of all entire functions having proximate order ðrÞ and type :
Strong proximate orderis a function ðrÞ 2 C2ðR
þÞrepresentable in the form
ðrÞ ¼ þ#2ðlog rÞ #1ðlog rÞ
log r , ð6Þ
where #1 and #2 are concave functions on R satisfying conditions (i ¼ 1, 2):
(i) limx!1#iðxÞ ¼ 1;
(ii) limx!1#iðxÞ=x ¼0;
(iii) limx!1#00iðxÞ=#0iðxÞ ¼0:
Note that these properties imply lim x!1# 0 iðxÞ ¼x!1lim # 00 iðxÞ ¼0, i ¼1, 2: ð7Þ
By Levin’s theorem ([4], p.39), any entire function of order has its own strong proximate order (6), that is such a strong proximate order ðrÞ ! that the type
:¼ lim sup
r!1
is finite and positive. We denote by ½ðrÞ, the set of all entire functions of strong
proximate order ðrÞand type :
Our first result is the following.
THEOREM1
(i) If f 2 ½ðrÞ, , then lim inf
jwj!1 jwj
ðjwjÞ1Rðw, f Þ ðe2Þ1: ð8Þ
(ii) If f 2 ½ðrÞ, has nonnegative Taylor coefficients, then lim inf
jwj!1jwj
ðjwjÞ1Rðjwj, f Þ ðeÞ1
: ð9Þ
(iii) There exists f 2 ½ðrÞ, such that
lim inf
jwj!1 jwj
ðjwjÞ1Rðw, f Þ ðeÞ1
: ð10Þ
We do not know whether the constants ðe2Þ1and ðeÞ1 in the right hand sides
of (8) and (9) are the best possible. Nevertheless, part (iii) of Theorem 1 shows that the best possible constant is not greater than ðeÞ1.
Let us compare Theorem 1 with Macintyre’s Theorem A. It is easy to see that, for f 2 ½ðrÞ, , Theorem A implies
(i0) For some sequence of w tending to 1,
Rðw, f Þ > Cjwj1ðjwjÞ=2: ð11Þ
(ii0) For jwj =2A
, where A Rþsatisfies (4),
Rðw, f Þ > Cjwj1ðjwjÞ=2: ð12Þ
Here C denotes a positive constant.
Part (i) of Theorem 1 shows that Rðw, f Þ > Cjwj1ðjwjÞ:This estimate is less precise than (11) and (12), but it is valid for all w. Moreover, part (iii) of Theorem 1 shows that Macintyre’s estimates (11) and (12) cannot be valid for all w.
If we consider functions of regular growth, then we can get a better bound for Rðw, f Þ, than (8).
THEOREM2 Let f 2 ½ðrÞ, . Assume
log Mðr, f Þ ¼ rðrÞþOð ðrÞÞ, r ! 1, ð13Þ where ðrÞ >0 is a nondecreasing function such that
ðiÞ ðrÞ ¼ oðrðrÞÞ, r ! 1,
then lim inf jwj!1 jwj ðjwjÞ1Rðw, f Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjwjÞ jwjðjwjÞ s >0: ð15Þ
It is easy to see that (15) is a better estimate than (8). Moreover, the smaller the func-tion is the better the bound for Rðw, f Þ is. In particular, if ðrÞ ¼ Oð1Þ, r ! 1, then Rðw, f Þ > Cjwj1ðjwjÞ=2, i.e. Macintyre’s bound (11) remains valid for all w in this case. In general, the bound (15) cannot be improved as the example of the Weierstrass sigma-function with lattice consisting of integer points of C shows. In this example, (13) holds with ðrÞ 2, ¼ =4, ðrÞ 1 (see [2], p.346) and, evidently,
Rðw, f Þ 1=pffiffiffi2.
3. PROOF OF THEOREM 1
Without loss of generality let us assume that f ð0Þ ¼ 1. We need the following lemma. LEMMA3.1 The following inequality holds
lim sup
r!1
Kðr, f Þ
rðrÞ e: ð16Þ
Proof For any k 2 ð1, 1Þ we have
log Mðkr, f Þ ¼ Zkr 0 d dtlog Mðt, f Þ dt ¼ Zkr 0 Kðt, f Þ t dt Zkr r Kðt, f Þ t dt Kðr, f Þ Zkr r dt t ¼Kðr, f Þ log k: Hence, lim sup r!1 Kðr, f Þ rðrÞ 1
log klim supr!1
log Mðkr, f Þ ðkrÞðkrÞ ðkrÞðkrÞ rðrÞ : Using (5), we get lim sup r!1 Kðr, f Þ rðrÞ k log k:
Taking minimum with respect to k > 1 in the right hand side, we obtain (16). g Now we will prove part (i) of Theorem 1. Let w be a maximum modulus point of f . Consider the function
wðzÞ:¼
f ðw þ zÞ f ðwÞ :
Let
Qðh, wÞ :¼ max
jzj hjwðzÞj:
Evidently, Qðh, wÞ > 1 for h > 0 because wð0Þ ¼ 1. Since j f ðw þ zÞj Mðjwj þ jzj, f Þ,
we have
Qðh, wÞ Mðjwj þ h, f Þ Mðjwj, f Þ : Hence,
log Qðh, wÞ log Mðjwj þ h, f Þ log Mðjwj, f Þ ¼ Zjwjþh jwj d dtlog Mðt, f Þ dt ¼ Zjwjþh jwj Kðt, f Þ t dt Kðjwj þ h, f Þ Zjwjþh jwj dt t ¼Kðjwj þ h, f Þ log 1 þ h jwj Kðjwj þ h, f Þ h jwj:
By Lemma 3.1, for each > 0 there exists rsuch that Kðr, f Þ ðe þ ÞrðrÞfor r > r.
Hence,
log Qðh, wÞ ðe þ Þðjwj þ hÞðjwjþhÞ h
jwj, for jwj > r: ð17Þ Following Macintyre [3] consider the function
wðzÞ:¼
QðwðzÞ 1Þ
Q2 wðzÞ
,
where Q ¼ Qðh, wÞ. Using properties of bilinear transformation and taking into account that by definition of Q, we hav e jwðzÞj Qfor jzj h, we conclude that j wðzÞj 1
for jzj h: Since wð0Þ ¼ 0, Schwarz lemma implies
j wðzÞj jzj=h, for jzj h: Hence, QjwðzÞ 1j jzj h jQ 2 wðzÞj jzj h ðjQ 21j þ j wðzÞ 1jÞ: Thus, jwðzÞ 1j ðjzj=hÞ ðQ21Þ Q jzj=h :
Since ðjzj=hÞ ðQ21Þ Q jzj=h <1, for jzj < h=Q, we get jwðzÞ 1j < 1, for jzj < h=Q: Hence, wðzÞ 6¼0, for jzj < h=Q,
and therefore f ðw þ zÞ 6¼ 0. This implies
Rðw, f Þ h=Q: Using (17), we obtain Rðw, f Þ h exp ðe þ Þ h jwjðjwj þ hÞ ðjwjþhÞ , jwj > r: Setting h ¼ ðeÞ1jwj1ðjwjÞ, we get Rðw, f Þ jwj 1ðjwjÞ e exp ðe þ Þ e ðjwj þ hÞðjwjþhÞ jwjðjwjÞ ( ) : Hence, lim inf jwj!1 jwj ðjwjÞ1Rðw, f Þ lim inf jwj!1 1 eexp ðe þ Þ e ðjwj þ hÞðjwjþhÞ jwjðjwjÞ ( ) : ð18Þ
Since (5) holds uniformly in k on any interval 0 < a k b < 1, we obtain
lim jwj!1 ðjwj þ hÞðjwjþhÞ jwjðjwjÞ ¼jwj!1lim ðjwjð1 þ h=jwjÞÞðjwjð1þh=jwjÞÞ jwjðjwjÞ ¼1:
Thus, inequality (18) reduces to
lim inf jwj!1 jwj ðjwjÞ1Rðw, f Þ 1 eexp 1 þ e :
Proof of (ii) Let f 6 0,
f ðzÞ ¼X
1
k¼0
akzk, ak0, k ¼ 0, 1, . . .
Then the set of maximum modulus points contains Rþ. If it does not coincide with Rþ,
then there is w =2 Rþ such that
X1 k¼0 akwk ¼ X1 k¼0 akjwjk:
This equality may hold if and only if
ak6¼0 ) argðwkÞ 0 ðmod 2Þ:
It follows that there is an integer n 2 such that ak¼0 for each k being not integer
multiple of n. Let us take the largest n with this property. Then we have f ðze2i=nÞ ¼
f ðzÞ and the set of maximum modulus points consists of the system of rays fz: arg z ¼ 2j=ng, j ¼ 0, . . . , n 1: Therefore without loss of generality we can con-sider further only maximum modulus points w lying on Rþ.
Let z ¼ rei’r, j’ rj< , be a zero of f . We hav e f ðrÞ ¼ f ðrÞ f ðrei’rÞ ¼X 1 k¼0 akrkð1 eik’rÞ j’rj X1 k¼0 kakrk¼ j’rjrf0ðrÞ: Whence j’rj f ðrÞ rf0ðrÞ¼ 1 Kðr, f Þ:
Assume that (9) is wrong. Then there exists a sequence 0 < wk! 1and a number
>0 such that
Rðwk, f Þ ðe þ Þ1w1ðwk kÞ ¼oðwkÞ, k ! 1: ð19Þ
By the definition of R, there are zeros zk¼rkei’k of f such that Rðwk, f Þ ¼ jzkwkj:
Note that (19) implies rk¼ ð1 þ oð1ÞÞwk, ’k¼oð1Þ: Hence,
Rðwk, f Þ j=zkj ¼rkj’kjð1 þ oð1ÞÞ
wk
Kðrk, f Þ
ð1 þ oð1ÞÞ:
This and (19) imply
Kðrk, f Þ
wk
Rðwk, f Þ
ð1 þ oð1ÞÞ ðe þ ÞwðwkÞ
Hence,
lim sup
r!1
Kðr, f Þ
rðrÞ ðe þ Þlim sup k!1 wðwkÞ k rðrkÞ k : ð20Þ
Since wk¼rkð1 þ oð1ÞÞ, (5) implies that the right hand side of (20) equals e þ
and we obtain a contradiction with (16). g Proof of (iii) Set
f ðzÞ ¼Y 1 k¼1 1 þ z pk apðpkÞk 0 @ 1 A, ð21Þ where pk¼e2 k , a ¼ e: We should show: (a) f 2 ½ðrÞ,
(b) lim infjwj!1jwjðjwjÞ1Rðw, f Þ ðeÞ1:
Let us first show (b). It is evident from definition (21) that f has nonnegative Taylor coefficients. Hence Rþ consists of maximum modulus points of f and in particular,
each point pkis such a point. Zeros of f are located at the points,
pkexp ið1 þ 2jÞ= apðpk kÞ h i n o , j ¼0, 1, . . . , apðpkÞ k h i 1, k ¼ 1, 2, . . . Evidently, Rðpk, f Þ pkpkexp i= apðpk Þ k h i n o ¼ 2pksin 2 apðpkÞ k h i : Therefore, pðpkÞ1 k Rðpk, f Þ 2pðpk kÞsin 2 apðpkÞ k h i :
Taking limit as k ! 1 and remembering that a ¼ e, we obtain (b). For part (a), we need to show that
lim sup
r!1
rðrÞlog Mðr, f Þ ¼ : ð22Þ
Let us first prove that the following relation holds
log Mðr, f Þ ¼ apðpnÞ n log r pn þoðrðrÞÞ, r ! 1, pn< r pnþ1: ð23Þ
We have log Mðr, f Þ ¼X 1 k¼1 log 1 þ r pk apðpkÞk 0 @ 1 A ¼ apðpnÞ n log r pn þ X n1 k¼1 apðpkÞ n log r pk þX n k¼1 log 1 þ pk r apðpkÞn ! þ X 1 k¼nþ1 log 1 þ r pk apðpkÞn 0 @ 1 A ¼: apðpnÞ n log r pn þS1þS2þS3:
It is easy to show that
S1¼o rðrÞ
, S2¼Oð1Þ, S3¼Oð1Þ, r ! 1, pn< r pnþ1,
which proves (23); we use that ðrÞ ! > 0 as r ! 1 (details omitted). Now let us prove (22), or equivalently,
lim sup
r!1
rðrÞlog Mðr, f Þ ¼ a=e: ð24Þ
Denoting the left hand side of (24) by and using (23), we obtain
¼ lim sup r!1 r2ðpn, pnþ1 rðrÞ apðpnÞ n logðr=pnÞ þoðrðrÞÞ lim sup r!1 r2ðpn, pnþ1 rðrÞ apðpnÞ n logðr=pnÞ alim sup n!1 pðpnÞ n max r2ðpn, pnþ1 gnðrÞ ð25Þ where gnðrÞ ¼ rðrÞlogðr=pnÞ.
Let us find an upper bound of gnfor pn< r pnþ1. We hav e
g0 nðrÞ ¼ r
ðrÞ1ðr0ðrÞlog r þ ðrÞÞ logðr=p nÞ þ1
½ :
Since r0ðrÞlog r þ ðrÞ ! as r ! 1, we have for large n
rðrÞ1h ð þ Þlogðr=pnÞ þ1 i < g0 nðrÞ < r ðrÞ1h ð Þlogðr=p nÞ þ1 i , ð26Þ
where is an arbitrary number from ð0, Þ.
When r < pne1=ðþÞ, the left inequality in (26) implies that g0nðrÞ >0. When
maximum value on ðpn, pnþ1at some point rn2 ½pne1=ðþÞ, pne1=ðÞ:Therefore, using (5) and (25), we obtain alim sup n!1 pðpnÞ n gnðrnÞ ¼alim sup n!1 pðpnÞ n logðrn=pnÞ rðrnÞ n alim sup n!1 pðpkÞ n ðpne1=ðþÞÞðpne 1=ðþÞÞ 1 ¼ a e=ðþÞ 1 ð Þ: Letting ! 0, we get a=e:
For the reverse inequality, let sn¼pne1=. Since limr!1pnþ1=pn ¼ 1, we have
pn< snpnþ1 for sufficiently large n. Therefore, using (5), we obtain
lim sup n!1 ðapðpnÞ n 1Þ logðsn=pnÞ rðrnÞ n ¼alim sup n!1 pðpnÞ n ðpne1=Þðpne 1=Þ 1 ¼ a e: Hence, (24) is true. g 4. PROOF OF THEOREM 2 Let us denote LðrÞ ¼ rðrÞ: ð27Þ By (6) we have
LðrÞ ¼expf#2ðlog rÞ #1ðlog rÞg:
We need the following amplification of (5) for a strong proximate order. LEMMA4.1 The following relation holds
ðkrÞðkrÞrðrÞ¼rðrÞðk1Þ þ kðk 1Þrþ1L0ðrÞ
þkðk 1Þ2rðrÞoð1Þ, r ! 1, ð28Þ
uniformly in each interval0 < a k b < 1. Proof We have ðkrÞðkrÞrðrÞ¼rLðrÞðk1Þ þ ðkrÞ½LðkrÞ LðrÞ: By Taylor’s formula, jLðkrÞ LðrÞ ðk 1ÞrL0ðrÞj ¼1 2ðk 1Þ 2 r2jL00ðcÞj, ð29Þ
for some c between r and kr. Since c2L00ðcÞ ¼ LðcÞn½#0 2ðlog cÞ # 0 1ðlog cÞ 2þ ½#00 2ðlog cÞ # 00 1ðlog cÞ ½#02ðlog cÞ #01ðlog cÞo,
using (7) we obtain (without loss of generality we assume a < 1 < b)
max
ar c brc
2jL00ðcÞ=LðcÞj !0, as r ! 1:
Taking into account that LðcÞ=LðrÞ ! 1, as r ! 1 uniformly in c, ar c br, we get
max ar c br r2L00ðcÞ LðrÞ ¼ar c brmax r2 c2 c2L00ðcÞ LðcÞ LðcÞ LðrÞ!0, as r ! 1,
and using (27) and (29) we get (28). g We also need the following amplification of (16) for functions satisfying conditions of Theorem 2.
LEMMA4.2 If(13) and (14) is satisfied, then
Kðr, f Þ ¼ rðrÞþrþ1L0ðrÞ þ O ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðrÞ
ðrÞ
p
, r ! 1:
Proof Let 1 k 3. Using the conditions (13) and (14), we get ½logMðkr,f Þ logMðr,f Þ ½ðkrÞðkrÞ rðrÞ ¼ Oð ðkrÞÞ Oð ðrÞÞ C ðrÞ, ð30Þ where C does not depend on k or r. Since
log Mðkr, f Þ log Mðr, f Þ ¼ Z kr r d dt log Mðt, f Þ dt ¼ Zkr r Kðt, f Þ t dt Kðr, f Þ Zkr r dt t ¼Kðr, f Þ log k, we obtain by (30) Kðr, f Þ 1 log k ðkrÞ ðkrÞ rðrÞþC ðrÞ :
Using Lemma 4.1, we get (uniformly in k 2 ½1, 3) Kðr, f Þ 1 log kr ðrÞ ðk1Þ þ 1 log kk ðk 1Þrþ1L0ðrÞ þ 1 log kk ðk 1Þ2rðrÞ oð1Þ þ 1 log kC ðrÞ: ð31Þ Using log Mðr, f Þ log Mðr=k, f Þ ¼ Zr r=k Kðt, f Þ t dt Kðr, f Þ log k,
Lemma 4.1 and (30), we obtain
Kðr, f Þ 1 log kr ðrÞ ð1 kÞ þ 1 log kk ð1 k1Þrþ1L0ðrÞ þ 1 log kk ð1 k1Þ2rðrÞ oð1Þ 1 log kC ðrÞ: ð32Þ Choosing k ¼1 þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrÞrðrÞ ,
we see that the estimate in (31) will give the upper bound and the estimate in (32) will give the lower bound for Kðr, f Þ. g Now we will prove Theorem 2. Let w be a maximum modulus point. Following Macintyre [3] define wðzÞ:¼ f ðwezÞ f ðwÞ e Kðjwj, f Þz: Let Pðh, wÞ :¼ max jzjhjwðzÞj, 0 < h 1 : Denoting jwj by r, we have
log jwðzÞj log MðreRe z, f Þ log Mðr, f Þ Kðr, f ÞRe z:
Set Re z ¼ t. Then
log Pðh, wÞ max
h t h
It is easy to see that
log Mðret, f Þ log Mðr, f Þ tKðr, f Þ ¼ Zret r ½Kðu, f Þ Kðr, f Þdu u ¼ Zret r
½Kðu, f Þ uðuÞuþ1L0ðuÞ ½Kðr, f Þ rðrÞ
rþ1L0ðrÞdu u þ
Zret r
½uðuÞþuþ1L0ðuÞ ½rðrÞþrþ1L0ðrÞdu u : Using Lemma 4.2 and the identity
rd drr ðrÞ ¼rd drr LðrÞ ¼ rðrÞ þrþ1L0ðrÞ, we get
log Mðret, f Þ log Mðr, f Þ tKðr, f Þ
Zret r C1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðuÞ ðuÞ p du u þ Z ret r C1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p du u þ ðretÞ ðretÞ rðrÞt½rðrÞþrþ1L0ðrÞ ¼: Y1þY2þY3: ð34Þ
Let us estimate Y1, Y2, Y3. Since r ðrÞ
ðrÞ is monotonically increasing for sufficiently large r, we have for t < 0,
Y1C1jtj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p : Using (5) and (14), we have for 0 < t h < 1,
Y1C1t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðretÞðretÞ ðretÞ q C2t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p : Hence Y1C3jtj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p , for jtj h: ð35Þ Evidently, Y2¼C1jtj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p : ð36Þ
Finally using Lemma 4.1, we get
Y3¼
rðrÞ
ðet1Þ þ etðet1Þrþ1L0ðrÞ þ rðrÞetðet1Þ2oð1Þ trðrÞtrþ1L0ðrÞ
¼rðrÞ
Since oð1Þ’s are uniform with respect to t, jtj 1, we obtain
Y3C4r
ðrÞt2, ð37Þ
where C4>0 does not depend on both r and t.
Hence, by (34) and (35)–(37) we conclude
log Mðret, f Þ log Mðr, f Þ tKðr, f Þ D1jtj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p þD2r ðrÞt2: Using (33) we obtain log Pðh, wÞ D1h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p þD2r ðrÞ h2: ð38Þ Introducing wðzÞ:¼ PðwðzÞ 1Þ P2 wðzÞ ,
we conclude similarly as we did for the function win the proof of Theorem 1.(i), that
wðzÞ 6¼0 for jzj < h=P:
Therefore
f ðwezÞ 6¼0 for jzj < h=P < 1: This implies (remind that r ¼ jwj)
Rðw, f Þ min jzj ¼ h=Pjw we zj ¼r min jzj¼h=Pj1 e zj rCh P:
Hence, with inequality (38) we obtain
Rðw, f Þ Crh exp n ðD1h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðrÞ ðrÞ p þD2h2r ðrÞÞo: Setting h ¼ 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðrÞ ðrÞ p
, we get for sufficiently large values of jwj
Rðw, f Þ C0jwj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 jwjðjwjÞ ðjwjÞ
p :
This is equivalent to (15). g
Acknowledgement
References
[1] A.A. Goldberg, B.Ya. Levin and I.V. Ostrovskii (1996). Entire and meromorphic functions. In: R. Gamkrelidze (Ed.), Encyclopedia of Math. Sciences, Vol. 85, pp. 1–193. Springer, Berlin.
[2] W.K. Hayman (1974). The local growth of power series: a survey of the Wiman-Valiron method. Canad. Math. Bull., 17, 317–358.
[3] A.J. Macintyre (1938). Wiman’s method and ‘‘flat regions’’ of integral functions. Quart. J. Math., 9, 81–88. [4] B.Ya. Levin (1980). Distribution of Zeros of Entire Functions. AMS, Providence, R.I.
[5] D. Ruelle (1969). Statistical Mechanics: Rigorous Results. Addison-Wesley, New York.
[6] Sh. Strelitz (1972). Asymptotic Properties of Analytical Solutions of Differential Equations. Mintis, Vilnius (Russian).
