Levy’s phenomenon for entire functions of several variables Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том б. № 2 (2014). С. 113-122.

УДК 517.55

LEVY’S PHENOMENON FOR ENTIRE FUNCTIONS OF

SEVERAL VARIABLES

A.O. KURYLIAK, O.B. SKASKIV, O.V. ZRUM

Abstract. For entire functions f (z) = ^+=o anzn, z £ C, P. Levy (1929) established that in the classical Wiman’s inequality Mf (r) ^ (r)(ln(r))1/2+e, e > 0, which holds

outside a set of finite logarithmic measure, the constant 1/2 can be replaced almost surely in some sense by 1/4; here M/(r) = max{|f (z)|: |z| = r}, /!/(r) = max{|an|rn: n > 0}, r > 0.

In this paper we prove that the phenomenon discovered by P. Levy holds also in the case of Wiman’s inequality for entire functions of several variables, which gives an affirmative answer to the question of A. A. Goldberg and M. M. Sheremeta (1996) on the possibility of this phenomenon.

Keywords: Levy’s phenomenon, random entire functions of several variables, Wiman’s inequality

Mathematics Subject Classification: 30B20, 30D20

1. Introduction

For an entire function of the form

/ (z) = ^ anZn n=0

we denote M/(r) = max{|/(z)|: |z| = r}, ^/(r) = max{|an|rn: n > 0}, r > 0. It is well known ([1], [2]) that for each nonconstant entire function / and all e > 0 the following inequality

Mf(r) ^ ^f(r)(ln^f(r))1/2+£ (1) holds for r > 1 outside an exceptional set Ef (e) of finite logarithmic measure (JE ^ dr < +ro). In this paper we consider entire functions of p complex variables

/ (z) = /(zb ...,zp) = ^ anZn

n|| = 0

where zn = zn1 ... zpn'p, p G N, n = (n1,..., np) G Z+, ||n|| = YJp=1 nj. For r = (r1,..., rp) G R+

we denote

B(R) = {t g R+: tj > Rj, j G {1,... ,p}}, R = (R1,..., Rp), ln2 x = lnlnx, rA = min ri, Mf (r) = max{|/(z)|: |z11 = Г1,... , |zp| = rp},

^f (r) = max{|an|rn1 ... r^ : n G Z+}, Mf (r) = ^ |an|r'

n =0

А.О. Курыляк, О.Б. Склскив, О.В. Зрум, Эффект Леви для целых функций многих переменных.

© А.О. КияуЫАК, О.В. Якаяюу, О.У. ZRUM 2014.

Поступила 7 октября 2013г.

By Ap we denote the class of entire functions of form (2) such that /(z) ^ 0 in Cp for any j £ {1,... ,p}. We say that a subset E of R+ is a set of asymptotically finite logarithmic measure [9] if E is Lebesgue measurable in R+ and there exists an R £ R+ such that E fl B (R) is a set of finite logarithmic measure, i.e.

/ ■■■ /n j < -

EnB(R) j 1

For entire functions of the form (2) analogues of inequality (1) are proved in [3, 5, 6, 9]. Also analogues of inequality (1) without exceptional sets for entire functions of several complex variables can be found in [10].

In particular, the following statement is proved in [9].

Theorem 1. Let / £ Ap and 5 > 0.

a) Then there exist R £ R+ and a subset E of B(R) of finite logarithmic measure such that for all r £ B (R)\E we have

/ P \ 1/2+5

Mf (r) ^ jf (r)(ninp-1 r* ■ lnp jUf (r)J . (3)

p— 1

in*

i=1

b) If for some a £ R+ we have M(r) > exp(ra) = exp^1 ... rP*p), as rA ^ or more

+

generally, for each 3 > 0

p

n dr*

*=1 < +ro, as SA ^ +ro,

J J r1r2 ... rp ln^ Mf (r)

B(S)

then there exist R £ R+ and a subset E of B(R) of finite logarithmic measure such that for all r £ B (R)\E we have

Mf (r) ^ jf (r) lnp/2+5 jf (r).

2. Wiman’s type inequality for random entire functions of several

variables

Let Q = [0,1] and P be the Lebesgue measure on R. We consider the Steinhaus probability space (Q, A, P) where A is the a-algebra of Lebesgue measurable subsets of Q. Let X = (Xn(t)) be some sequence of random variables defined in this space. For an entire function of the form /(z) = +=o anzn by K(/, X) we denote the class of random entire functions of the form

+^

/(z, t) = ^ anXn(t)zn (5)

n=o

In the sequel, the notion “almost surely” will be used in the sense that the corresponding property holds almost everywhere with respect to Lebesgue measure P on Q = [0,1]. We say that some relation holds almost surely in the class K(/, X) if it holds for each entire function /(z, t) of the form (5) almost surely in t.

In the case when R = (Xn(t)) is the Rademacher sequence, i.e. (Xn(t)) is a

sequence of independent uniformly distributed random variables on [0,1] such that P{t: Xn(t) = ±1} = 1/2, P. Levy [7] proved that for any entire function we can replace the constant 1/2 by 1/4 in the inequality (1) almost surely in the class K(/, R). Later P. Erdos and A. Renyi [8] proved the same result for the class K(/, H), where H = (e2n*Wn(t)) is the Steinhaus sequence, i.e. (wn(t)) is a sequence of independent uniformly distributed random variables on [0,1]. This statement is true also for any class K(/, X), where X = (Xn(t)) is multiplicative

system (MS) uniformly bounded by the number 1. That is for all n £ N and t £ [0,1] we have |Xn(t)| ^ 1 and

(V1 ^ i1 < i2 < ■ ■ ■ < ): M(X*iX*2 ■ ■ ■ X*fc) = 0,

where M£ is the expected value of a random variable £ ([15]—[16]).

In the spring of 1996 during the report of P. V. Filevych at the Lviv seminar of the theory of analytic functions professors A. A. Goldberg and M. M. Sheremeta posed the following question (see [12]). Does Levy’s effect take place for analogues of Wiman’s inequality for entire functions of several complex variables?

In the papers [12]—[14] we have found an affirmative answer to this question for Fenton’s inequality [4] for entire functions of two complex variables.

In this paper we will give answer to this question for Wiman’s type inequality from [9] for entire functions of several complex variables.

The exceptional set in our statements is “smaller” than the exceptional set in the corresponding theorems from [4], [12]—[14]. The method of proof in this paper differs from the method of the papers [4], [12]—[14].

Let Z = (Zn(t)) be a complex sequence of random variables Zn(t) = Xn(t) + iYn(t) such that both X = (Xn(t)) and Y = (Yn(t)) are real MS and K(/, Z) the class of random entire functions of the form

/(z, t) = ^ anZn(t)zni ... zn

n =o

Theorem 2. Let Z = (Zn(t)) be a MS uniformly bounded by the number 1, 5 > 0, / £ Ap.

a) Then almost surely in K(/, Z) there exist R £ R+ and a subset E* of B(R) of finite logarithmic measure such that for all r £ B(R)\E* we have

( 1 \ 1/4+5

m/(r,t) = max|/(z,t)| ^jf(r)(inPjf(r)ninP r*) . (6)

z =r V /

*=1

b) If for some a £ R+ we have

M(r) > exp(ra) = exp(rai... ra) as rA —

or more generally, for each 3 > 0 inequality (4) holds, then almost surely in K(/, Z) there exist R £ R+ and a subset E of B(R) of finite logarithmic measure such that for all r £ B (R)\E we get

Mf (r, t) ^ jf (r) lnp/4+5 jf (r). (7)

Lemma 1 ([10]). Let X = (Xn(t)) be a MS uniformly bounded by the number 1. Then for each 3 > 0 there exists a constant A^p > 0, which depends on p and 3 only such that for all N > N1(p) = max{p, 4n} and {cn: ||n|| ^ N} C C we have

N

P t : max

eini ^1 p*np^p

nXn( ^ e

n =o

: ^ £ [0, 2n]p >>

> A^pSN ln2 Nj ^ N_

where S2 = £* = |cn|2.

By H we denote the class of function h: R+ — R+ such that

^+ —— R+

f f du1... du

J J h(u)

11

p < +oo.

We also define for all i £ {1,... ,p}

d i +~

lnm/(r) =r*—(lnm/(r)) = m-n)^2 n*|ankn

dr* Mf (r) ||n||=0

Lemma 2 ([9]). Let h £ H. Then there exist R £ R+ and a subset E' of B(R) of finite

logarithmic measure such that for all r £ B(R)\E' and s £ {1,... ,p} we have

ln Mf (r) ^ h(ln r1,..., ln rs—1, ln Mf (r), ln rs+1,..., ln rp). (9)

Proof of Theorem 2. Without loss of generality we may suppose that Z = X = (Xn(t)) is a

MS. Indeed, if Zn(t) = Xn(t) + iYn(t) then we obtain

+ ^ +^

/(z,t)= ^ anXn(t)zn + ^ ianYn(t)zn = A(z,t) + /2(z,t),

| n| =o | n| =o

where /1,/2 £ K(/, X), and

max{ j(r, /1 (-,t)), j(r,/2(-,t))} ^ j(r,/) = max^nK1 ... r^ : n £ Z+}

for all r £ R+ and t £ [0,1]. Then from inequality (6) we obtain that there exists a set E0 of asymptotically finite logarithmic measure such that for all r £ B (R)\E0 almost surely in

K (/,Z)

1 \ 1/4+50

Mf.(r, t) ^ jf (r)(nlnp r* - lnp jf(r)J , j £ {1, 2}, 5o > 0.

*=1

So, for large enough RA and for all r £ B(R)\E0 almost surely in K(/, Z) we get

Mf(r,t) ^ Mfi(r,t) + Mf2(r,t) ^

/-1^1 \ 1/4+50 /'l^ 1 \ 1/4+250

^ 2jf (r^lnp r* - lnp jf (r)j < jf (r^lnp r* - lnp jf (r)

*=1 *=1 For any j £ {1,... , p} we have

lim jf(r0,...,r°—1,rj,r°+1,...,rp) =(10)

for fixed r0 > 0, i £ {1,... ,p}\{j}. Indeed, if (10) does not hold, then there exists a constant C > 0 such that for all rj > r* we have jf (r0,... , r0— 1, rj, rj+1,... , rp) < C < +ro. Hence, #{nj > 1: an = 0} = 0 and -f^/(z) = 0 in Cp. So, / £ Ap, which gives a contradiction.

For k £ N U {0} we denote Gk = {r = (r1,... , rp) £ R+: k ^ ln jf (r) < k + 1} If [1; +w)p. Then Gk = 0 for k > k0 and from (10) we deduce that for all k the set Gk is a bounded set. Let G+ = U +=k Gj and

p

h(r) = n r* ln1+* r* £ H, 51 > 0.

r*

*=1

By Lemma 2 there exist Rj £ R+ and a subset Ej of B (Rj) of finite logarithmic measure such that for all r £ B(Rj)\Ej and j £ {1,... ,p} we have

+

n*|an|rn ^ Mf (r)h(ln r1,..., ln rs—1, ln Mf (r), ln rs+1,..., ln rn) ^

| n| =o

p

^ Mf (r) ln Mf (r)ln2+‘Ji Mf (r) n ln r* ln2+* r*.

*=1, *=j

We can choose R £ R+ so that B(R) C ^Hp=1 B(Rj)^ f[ee2, +to)p

Then for for large enough RA and for all r £ B(R)\(up=1E*) we obtain

+00 p / p \

E |n||an|rn ^ Mf(r)ln Mf M1^1 Mf(r) E n ln r* ln2+‘Ji r* 1 ^

||n||=0 j=1 \*=1, *=j /

p

^ p - Mf (r)ln1+5i/2 Mf (r) n ln r* ln2+* r*,

*=1

By Theorem 1 we get for large enough RA and for all r £ B(R)\^p=1 E*)

+00 p

V—/T-f , \ 1/2+5l

||n|||an|rn ^ pjf(rmlnp r* - lnp jf(r)

||n|| =o *=1

x(ln jf (r) + (2 + 50 ((p - 1) E ln2 r* + p ln2 jf (r^j n ln r* ln^*

*=1 *=1

« (r)(ln (r))p/2+(p+.)5l + . (ft ln r*\(p—1^)(1/2+il) + 1 (JJ ln2 r*\2+Sil/2,

*=1 *=1

because a1x1 + - - - + akxk < x1 - ... - xk for large enough xA > 1, x = (x1,... , xk). Therefore as 52 = (p + 1)51 for large enough RA and for all r £ B(R)\(Up=1 E*) we obtain

+ p 1+^2

E iniKr ^ jf(r)lnp/2+1+52 jf(r) n( lnp r* ln2 r*^ .

||n|| =o *=1

So,

where

E K|rn ^ E ^|an|rn = d E l|n||K|rn ^

||n||>d ||n||>d ||n||>d

p / \

^ djf (r) lnp/2+1+52 jf (r) n (lnp r* ln2 r*) = jf (r), (11)

d i=1

d = d(r) = lnp/2+1+52 jf (r) n (lnp r* ln2 r*\ .

i=1

Let Gk = Gk \ Ep+1, Ep+1 = (Jp=1(E* U E*) U k=—1 G*^. By I we denote the set of integers

k > k0 such that Gk = 0. Then #1 = +ro. For k £ 1 we choose a sequence r(k) £ Gk. Then

for all r £ Gk we get

jf(r(k)) < ek+1 ^ ejf(r), jf(r) < ek+1 < ejf(r(k)), (12)

and also

+

J G* = J Gk \ Ep+1 = J Gk \ Ep+1 = [1; +^)p \ Ep+1.

ke/ ke/ k=1

For k £ 1 we denote Nk = [2d1(r(k))], where

p / \ d1(r) = lnp/2+1+52 (ejf (r)) n lnp r* ln2 r*

i=1

and for r £ Gk

WNk (r, t) = max

E «nr? ... rne*ni*+...+m^pXn(t)

l|n|KNfc

For a Lebesgue measurable set G C Gk and for k £ 1 we denote

measp(G)

: ^ £ [0, 2n]p

vk (G)

meaSp(Gk)

where measp denotes the Lebesgue measure on Rp. Note that vk is a probability measure defined on the family of Lebesgue measurable subsets of G*.

Let Q = (Jkef G* and 1 = {kj: j > 0}, where kj < kj+1, j > 0. Without loss of generality we may assume that k0 = 0. Then Ep+1 = (Jp=1(E* U E*). For Lebesgue measurable subsets G of Q we denote

+0 1 ( (1 \kj+i—kj \

v (G) = E 2k. I1 - (2) )- Vk.+i(G f Gkj+i).

j=o 2 2

13)

We note that vkj+l (Gk J = 1, therefore

+0 + 00 kj+i 1 + 00

'(Q) = E2.0 - /2) j+,— j\vk.+i(Gk,+,) = E E * = E1 = 1.

j=o

j=0 s=k. + 1

s=1

2s

Thus v is a probability measure, which is defined on measurable subsets of Q. On [0,1] x Q we define the probability measure P0 = P ® v, which is a direct product of the probability measures P and v. Now for k £ 1 we define

Fk = {(t, r) £ [0,1] x Q: WNfc(r, t) > AS^(r) ln1/2 Nk},

Fk(r) = {t £ [0,1]: WNk(r, t) > A1SN(r)ln1/2 Nk},

where SNfc (r) = ENk||=0 l«n|2r2n and Ap is the constant from Lemma 1 with 3 = 1. Using Fubini’s theorem and Lemma 1 with cn = anrn and 3 =1, we get for k £ 1

Po(Fk)

1

1

dP dv = P(Fk(r))dv ^ —v(Q) = —.

Nk

Nk

Fk (r)

Note that Nk > lnp/2+1 jf (r(k)) > k3/2. Therefore ^ke/ P0(Fk) ^ S+zO! k—3/2 < +to. By Borel-Cantelli’s lemma the infinite quantity of the events {Fk: k £ 1} may occur with probability zero. So,

+

Po(F) = 1, F = J p| Fk C [0,1] x Q.

s=1 k>s,ke/

Then for any point (t,r) £ F there exists k0 = k0(t,r) such that for all k > k0, k £ 1 we have

WNk (r,t) ^ ASNk (r)ln1/2 Nk. (14)

Let Pj be a probability measure defined on (Qj, Aj), where Aj is a a-algebra of subsets

Qj (j £ {1,...,p}) and P0 is the direct product of probability measures P1,...,Pp defined on (Q1 x ... x Qp, A1 x ... x Ap). Here A1 x ... x Ap is the a-algebra, which contains all A1 x ... x Ap, where Aj £ Aj .If F C A1 x ... x Ap such that P0(F) = 1, then in the case when projection

F1 = {t1 £ Q1 : (3(t2, . . . , tp) £ Q2 x ... x Qp)[(t1, ... , tp) £ F]}

of the set F on Q1 is P1-measurable we have P^i(F1) = 1.

By Fq we denote the projection of F on Q, i.e. Fq = {r £ Q: (3t)[(t,r) £ F]}. Then

v(Fq) = 1. Similarly, the projection of F on [0,1], F[01] = (Jreq F(r), we obtain P(F[01]) = 1.

Let FA(t) = {r £ Q : (t, r) £ F}. By Fubini’s theorem we have

0 = J(1 - Xf)dPo = J I J(1 — XfAW)dv jdP-X o ^ q '

So P-almost everywhere 0 = Jq(1 — XFA(t))dv =1 — v(FA(t)), i.e. 3 F1 C F[01], P(F1) = 1 such that for all t £ F1 we get v(FA(t)) = 1.

Indeed, if for some k £ 1, k = kj+1 we obtain vk(FA(t) If Gk) = q < 1, then

a x—v a +° 1 ( (1 \ks+i—ks \

v(FA(*)) = E vk(FA«) f Gk) < E 1 — (2) ) —

ke/ s=0

. 1 ( (1 \k7+l—k7 \ . ,1 ( (1 \k7+l—k7 \

—(i—9)2*7 (!—y )= 1—(1—('—G) ) <L

For any t £ F1 and k £ 1 we choose a point r0k) (t) £ Gk such that

WN (r0k)(t),t) > 4 Mk(t), Mk(t) d= SUp{WNfc (r,t) : r £ Gk}.

Then from vk(FA(t)flGk) = 1 for all k £ 1 it follows that there exists a point r(k)(t) £ Gk fFA (t) such that

WN(r0k)(t),t) — Wn.(r(k)(t),t)| < 4Mk(t)

or

3 1

4Mk(t) « W'N(r‘k)(t),f) « WNk(r(k'){(},() + 4Mk(t).

Since (t,r(k)(t)) £ F, from inequality (13) we obtain

1 Mk(t) ^ WNk(r(k)(t),t) ^ A1SNk(r(k)(t))ln1/2 Nk. (15)

Now for r(k) = r(k)(t) we get

S2(r(k)) ^ jf(r(k))Mf(r(k)) ^ jf(r(k))/nlnp—1 r(k) - lnp jf(r(k))\1/2+5.

*=1

So, for t £ F1 and all k > k0(t), k £ 1 we obtain

Sn(r(k)) ^ jf (r(k))(nlnp—1 r(k) - lnp jf (r(k))\1/4+5/2. (16)

*=1

It follows from (12) that d1 (r(k)) > d(r) for r £ Gk. Then for t £ F1, r £ FA(t) f Gk, k £ 1, k > k0(t) we get

Mf(r,t) ^ E l«n|rn + WNk(r,t) ^ E l«n|rn + Mk(t).

I|n||>2dl (r(k)) l|n||>2d(r)

Finally, from (11), (15), (16) for t £ F1, r £ FA(t) f Gk, k £ 1 and k > k0(t) we deduce

Mf (r(k) ,t) ^ jf (r(k)) + 2ApSNk (r(k)) ln1/2 Nk ^

£ jf (r(k)) + 2Apjf (r(k))(jlnp—1 r(k) - lnp jf (r(k))\1/4+5/2x

*=1

p

x ^ (p/2 + 1 + 52) ln2(ejf (r(k))) + (1 + 52) E(p ln2 r*(k) + 2ln3 r(k))j

*=1

Using inequality (12) we get for t £ F1, r £ FA(t) f G*, k £ 1 and k > k0(t)

1/4+352/4

1/2

/-r-T 1 \ 1/4+302/4

Mf(r,t) ^ Cjf(r)(nlnp r* •lnp jf(r)) . (17)

p— 1

ln

*=1

We choose k1 > k0(t) such that for all r £ G+ we have

02/4

C ^ lnp 1 r* - lnp jf (r) J . (18)

*=1

Using (17) and (18) we get that inequality (6) holds almost surely (t £ F1, P(F1) = 1) for all

r £ (J (Gk n F ''(t» n G+,\\E * =

= ([1, +rc)p f G+,) \ (E* U G* U Ep+0 = [1, +rc)p \ Ep+2,

ke/ k+i

where

Ep+2 = Ep+1 U G* U E*, G* = J(Gk \ FA(t)).

ke/

It remains to remark that v(G*) defined in (13) satisfies v(G*) = ^2ke/(vk(Gk) — vk(FA(t))) = 0. Then for all k £ 1 we obtain

vk(g* \ fAffl=m^« = 0,

measp(Gk)

meaSp(Gk \ FA(t)) = f - - - f dVl'.' dVp =0. □

J J r1... rp

Gk\F A(t)

3. Some examples

In this section we prove that the exponent p/4 + 5 in the inequality (7) cannot be replaced by a number smaller than p/4. It follows from such a statement.

Theorem 3. For /(z) = exp{J^p=1 z*} almost surely in K(/, H) for r £ E we have

Mf(r,t) > jf(r)lnp/4 jf(r),

where E is a set of infinite asymptotically logarithmic measure and H = {e2n*Wn}, {wn} is a sequence of independent random variables uniformly distributed on [0,1].

In order to prove this theorem we need such a result.

Theorem 4 ([17]). For the entire function g(z) = ez almost surely in K(g, H) we have

Mg(r,t) ^ ^

lim -------^-------- >W —. (19)

w + 0 jg (r)ln/jg (r) V 8

Proof of Theorem 3. For the entire function /(z) = exp{J^p=1 z*} we have lnMf (r) = 5^= r* and for each 3 > 0 we get

dr1 . . . drp

11 1 p < +00.

J J r\ ... rp(r 1 + ... + rp)^

(1,+oo )p

Therefore the function /(z) satisfies condition (4). From (19) we have for r £ (r0, +to)p

1p

Mf(r,t) > jf (r)IIln1/4 jg (r*).

2p

*=1

Denote ^(r) = ln jg(r). Remark that

At = {r: n = t; r* £ (t1,t2) = (^—1(^(r1)/2),^—1 (2^1)))} C

p 1 p p

C {r: n >Hr*} > T^Tp (E ^r*}'p

*=1 v *=1

Indeed, if r £ At then for fixed r1 we obtain

II ^(r*) = j] «r*} > ^r,} I] ^ ^ =

*=1 *=2 *=2

1 1 \p

= 2p—1(2p — 1)p (^(r1) + 2^(r1) +... + 2^(r1))p > (4p^U^ ^(r*V .

For r £ A = U +Oro At we get

1 p 1 / p \ p/4

Mf(r,t) > 2p jf (r^ln1/4 jg(r*) > jf (r)78p^(5] ln jgWJ >

*=1 ( p) *=1

> (8p)p jf(r)lnp/4 jf(r).

It remains to prove that the set A has infinite asymptotically logarithmic measure. It is known [11] that t < x\)—1 (t) < 3t/2, t — +to. Therefore,

+CO t2 t2 + CO / t2 \ p—1

"'■p«-/ /■/“-//f dri=

r0 ti ti r0 ti

+ CO

.p—

2 r1

r0

+ CO +co

> + (to^r,}} — ln/\ dl = lnp—. 8 - + ^ = +M. n

r0 r0

Acknowledgments. Authors are grateful to the Referee for their valuable suggestions and comments.

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Andriy O. Kuryliak,

Department of Mechanics and Mathematics,

Ivan Franko National University of L’viv,

Universytets’ka str. 1,

79000, Lviv, Ukraine E-mail: [email protected]

Oleh B. Skaskiv,

Department of Mechanics and Mathematics,

Ivan Franko National University of L’viv,

Universytets’ka str. 1,

79000, Lviv, Ukraine

E-mail: [email protected]

Oleh V. Zrum,

Department of Mechanics and Mathematics,

Ivan Franko National University of L’viv,

Universytets’ka str. 1,

79000, Lviv, Ukraine