Minimum modulus of lacunary power series and h-measure of exceptional sets Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 9. № 4 (2017). С. 137-146.

УДК 517.576

MINIMUM MODULUS OF LACUNARY POWER SERIES AND ^-MEASURE OF EXCEPTIONAL SETS

T.M. SALO, O.B. SKASKIV

Abstract. We consider some generalizations of Fenton theorem for the entire functions represented by lacunarv power series. Let f (z) = +=o Лz""fc> where (nk) is a strictly increasing sequence of non-negative integers. We denote by

Mf (r) = max{|/(z)|: |z| = r},

mf (r) = min{|/(z)|: |z| = r},

цf (r) = max{| Д : к ^ 0}

the maximum modulus, the minimum modulus and the maximum term of f, respectively. Let h(r) be a positive continuous function increasing to infinity on [1, with a non-decreasing derivative. For a measurable set E С [1, we introduce h — meas (E) = JE . In this paper we establish conditions guaranteeing that the relations

Mf (r) = (1 + o(1))m/(r), Mf (r) = (1 + o(1))^/(r)

are true as г ^ outside some exceptional set E such that h — meas (E) < For some subclasses we obtain necessary and sufficient conditions. We also provide similar results for entire Dirichlet series.

Keywords: lacunarv power series, minimum modulus, maximum modulus, maximal term, entire Dirichlet series, exceptional set, h-measure

Mathematics Subject Classification: 30B50

1. Introduction

Let L be the class of positive continuous functions increasing to infinity on [0; By L+ we denote the subclass of L consisting of the diiferentiable functions with a non-decreasing derivative, and L~ stands for the subclass of functions with a non-increasing derivative. Let f be an entire function of the form

f (z) = £ fkzn* , (1)

k=0

where (nk) is a strictly increasing of nonnegative integers. Given r > 0, we denote by

Mf (r) = max{|/(z)|: |z| = r}, mf (r) = min{|/(z)|: |z| = r}, ^f (r) = max{|/fc|rrafc: к ^ 0} the maximum modulus, the minimum modulus and the maximum term of f, respectively. P.C. Fenton [1] (see also [2]) proved the following statement.

Theorem 1 ([1]). If

< (2)

Е-

= пм - m

t.m. salo, o.b. skaskiv,the minimum modulus of lacunary power series and h-measure of exceptional sets.

© Salo T.M., Skaskiv O.B. 2017. Поступила 22 июля 2016 г.

then for every entire function f of the form (!) there exists a set E C [1, of finite

logarithmic measure, i.e. log-meas E := fE dlogr < such, that the relations

Mf (r) = (1 + o(1))m/ (r), Mf (r) = (1 + o(1))^ (r) (3)

hold as r ^ (r E E).

P. Erdos and A.J, Maeintvre [2] proved that condition (2) implies that (3) holds as r = rj ^ for some sequenee (rj),

Denote by D(A) the class of entire (absolutely convergent in the complex plane) Diriehlet series of the form

F(z) = ^ anezX", (4)

n=0

where A = (A^) is a fixed sequence such that 0 = A0 < \n t (1 ^ n t +rc>). Let us introduce some notations. Given F E x E R, we denote by

y,(x, F) = max{|ara|e^An : n ^ 0}

the maximal term of series (4), by

M(x, F) = sup{|F(x + iy)l: y E R}

we denote the maximum modulus of series (4), by

m(x, F) = inf{IF(x + iy)l: y E R}

we denote the minimum modulus of series (4), and

v(x, F) = max{n: |an|exA" = y(x, F)}

stands for the central index of series (4),

In [3] (see also [4]) we find the following theorem.

Theorem 2 ([3]). For every entire function F E D(A) the relation

F (x + iy) = (1 + o(1))a„ (5)

holds as x ^ outside some set E of finite Lebesgue mea sure (JE dx < uniformly in y eR, if and only if

1

^ A-T < (6)

1 — A

ra=0

Note, that in the paper [5] there were proved the analogues of other statements in the paper by P.C, Fenton [1] for subclasses of functions F E D(A) defined by various restrictions on the growth rate of the maximal term ^(x,F),

The finiteness of Lebesgue measure of an exceptional set E in theorem A is the best possible description. This is implied by the next statement.

Theorem 3 ([6]). For every sequence A = (Xk) (including those which satisfy (6)J and for every continuously differentiable function h: [0, ^ (0, such that h'(x) ^ (x ^ +<x>) there exist an entire Dirichlet series F E D(\), a constant [3 > 0 and a measurable

set E1 C [0, +<x>) of infinite h-measu,re (h — meas (E1) = dh(x) = such that

(V xeE 1): M (x,F ) > (1 + ¡3)^(x,F), M (x,F) > (1 + ¡3 )m(x,F). (7)

Recently, Ya.V, Mykvtvuk remarked that in Theorem 3, it is sufficient to assume that a

h

h(x)

—> as x —> +oo.

It follows from Theorem 3 that the finiteness of logarithmic measure of an exceptional set E in Fenton's Theorem 1 is also the best possible description.

It is easy to see that the relation

F (x + iy) = (1 + o(1)K (x,F)e(x+iy)x"^F) holds as x ^ (x e E) uniformly in y e R if and only if

M(x,F) ~ ^(x,F) and M(x,F) ~ m(x,F) (x ^ x / E). (8)

In view of Theorem 3, the natural question arises: what conditions should an entire Dirichlet series satisfy in order to relation (5) be true as x ^ outside some set E2 ofhnite h-measure, i.e.,

h — meas (E2) < In this paper we provide the answer to this question as h e L+.

h—

According to Theorem 3, in the case h e L+, condition (6) must be fulfilled. Therefore, in the subclass

D(A, $) = {F e D(A) : lnF) ^ x$(x) (x>Xo)}, $ e L, it should be strengthened. The following theorem indicates this.

Theorem 4. Let $ e L,h e L+ and ip be the inverse function for the function $. If

+ ^0

(V6> 0): £ . 1 . tiL(\k)+ . b ,)< +«>, (9)

— Ah+1 — Ah V Ah+1 — Ak'

k=0

Afc+i — Xk \ Afc+i — Xk

then for all F e D(A, $) identity (5) is true as x ^ outside some set E of a finite h-measure uniformly in y e R.

Before proving this theorem, we need additional notations and an auxiliary lemma. Denote A0 = 0 and

n-1 rc , 1 1 x

A = — A,) £ [x _x + --— .

=0 _ +i ^m— 1 Am+1 ^m J

for n ^ 1, The next lemma is similar to Lemma 1 in [8], Lemma 1. For all n ^ 0 and k ^ 1, the inequality

is true, where an = eqAn, q > 0, and

, v . Q Ak-1 — aa Tk = rk (q) = qxk + T-7-, xk

Xk — Xk-1 Xk — Xk-1

Proof. Since

ln an — ln an-1 = q(An — An-1) = —qxn(Xn — Xn-1),

for n ^ k + 1 we have

ln — + Tk(\n - \k) = - q x3(X3 - X3-i) + rk (Xj - Xj-i )

ak j=k+i j=k+i

n

= - Y1 (qxi - Tk) (X3 - Xi-i) j=k+i n

j=k+i n

Q 1 = -q(n - k)-j=k+i

Similarly, for n ^ k - 1 we obtain

ln — + rk(\n - \k) = - ln — - rk(\k - Xn)

ak an

k

--Q

Y^ Xi (Xi - X3-i) - Tk Y (Xi - X3-i)

j=n+i

j=n+i k

= - Y1(rk - qxi) (xi- Xi-i)

j=n+i k

^ - Y (ri - qxi)(Xi - Xi-i)

j=n+i

k

qY 1 = -Q(k -

j=n+i

and this completes the proof.

□

a

an

n ezXn.

Proof of Theorem 4- We first note that condition (9) implies the convergence of series (6). We consider the function

f* (*) = £■

n=0

Since An ^ 0, we have fq G D(A) and v(x, fq) ^ (x ^ +œ).

Let J be the range of the central index v(x,fq). Denote by (Rk) the sequence of the jump points of central index, numbered in such a way that v(x,fq) = k for all x G [Rk, Rk+i) and Rk < Rk+i. Then for all x G [Rk, Rk+i) and n ^ 0 we have

a

—exXn e

According to Lemma 1, for x G [Rk + rk, Rk+i + rk) we obtain

anexXn . a.

ak

x\k

akexX* ak

^ ^ erk(An-Afc) ^ g-^^ (n ^ o).

Therefore, and

v(x,F) = k, ß(x,F) = akexXk (x G [Rk + rk,Rk+i + rk)) IF(x + iy) - av{XtF)e(x+w)K(*>F) | ^ ^ ß(x, F)e-^-^>F)|

n=v(x,F )

,-q

(11)

C2

1 - e—

ß(x,F)

for all x e [Rk + Tk, Rk+1 + rt^d k e J. Thus, inequality (12) holds for all

def +°°

x e E1(q) = U [Rk+1 + Tk, Rk+1 + Tk+1). k=0

Since

2q

Tk+1 — Tk = T-7-,

Ak+1 — Ak

and by the Lagrange theorem

h( Rk+1 + Tk+1) — h( Rk+1 + Tk) = (Tk+1 — Tk )h'( Rk+1 + Tk + ^ (rk+1 — Tk)), where 6k e (0; 1), for each q > 0 we have

rRk+1+ rk+i

h — meas (E1(q)) = ^^ j dh(x)

k=0 k+i +Tk +<x

= ^(h( Rk+1 + Tk+1) — h( Rk+1 + Tk)) (13)

k=0

+<x

^2q V --tíÍRk+i + Tk + 2g--).

— Ak+1 — Ak V Ak+1 — Ak '

k=0

Here we have employed the condition h e L+. For F e D(A, x > max{x0,1} we have

X

x$(x) ^ \n^(x,F) = ln^(1,F) + J )dx ^ ln^(1,F) + (x — 1)Xu(x-0,f).

1

This implies

X$(X) ^ X\v(x-0,F) (14)

for all x ^ x1 ^ x0, i.e.

X (Xu(X-0,F)) (X ^ X1).

Thus, according to (11), for k ^ k0 we obtain

Rk+1 + Tk ^ (Xv(Rk+1+Tk-0,F)) = ). Applying this inequality to inequality (13), by the condition h e L+ we have

1 1

h — meas(E1(q)) ^ 2 q V---h' L(Xk) + 2q---V (15)

Ak+1 — Ak V Ak+1 — Ak^

Therefore, using (9) we conclude that h — meas ( E1(q)) < Let qk = k. Since h — meas (E1(qk)) < we have

h — meas (E1(qk) H [x, = o(1) (x ^

hence, it is possible to choose an increasing to sequence (xk) such that

h — meas (£1(qk) H [xk; +rc>)) ^ jt.

k2

+00

for all k ^ 1. Denote E1 = (J {E1(qk) H [xk;xk+1)). Then

k=1

+0 +0 1

h - meas (Ei) = ^ h - meas (Ei(qk) H [xk; Xk+1)) ^ ^ ~¡2 < k=1 k=1

On the other hand, by inequality (12), for x G [xk; xu+i) \ El we get

\F(x + iy) — av(x>F| ^ 2 ---»(x, F),

1 — e qk

and therefore, as x ^ (x G El), we obtain (5), The proof is complete, □

We observe that if h(x) = x, then condition (9) becomes condition (6), and h-measure of the set E is its Lebesgue measure. Let $ G L. Consider the classes

Do(A, $) = {F G D(A) : (3K > 0)[ln¡i(x, $) ^ Kx$(x) (x > a^)]}, Di(A, $) = {F G D(A) : (3Ki,K2 > 0)[ln¡i(x, $) ^ Kix$(K2x) (x > a*)]}.

Theorem 5. Let $0 G L, h G L+ and tp0 be the inverse function for the function $0. If

1 / h \

(Vb > 0) : V ---h' MbK) + t-r < (16)

^=0 ^n+l — An \ An+i — AnJ

then for each function F G D0(A, $0) relation (5) holds as x ^ outside some set E of finite h - measure uniformly in y G R.

Theorem 6. Let $l G L, h G L+, and <pl be the inverse function to the function If

+ ^0

> 0)^ h'(b^i(b\„,)) < (17) „ An+i — An

n=0

then for every function F G Dl(A, $l) relation (5) holds as x ^ outside some set E of finite h-measure uniformly in y G R.

Proof of Theorems 5 and 6. Theorems 5 and 6 are implied immediately by Theorem 4,

Indeed, if F G D0(A, $0), then F G D(A, $) as $(x) = K$0(x). But in this case ip(x) = <p0(x/K) and condition (9) follows condition (16), Then it remains to apply Theorem 4.

In the same way, if F G Dl(A, then F G D(A, $) as $(x) = Kl$l(K2x). But in this case ip(x) = tpl(x/Kl)/K2 and hence, condition (9) follows condition (17), It remains to employ Theorem 4 once again, □

Remark 1. It is easy to see that for each, fixed functions h G L+ and $ G L there exists a A

The next theorem shows that condition (17) is necessary for relations (5), (8) to hold for each F G Dl(A, $l) as x ^ outside a set of a finite h-measure. Here we assume that condition (6) is satisfied.

Theorem 7. Let $l G L, h G L+, and tpl be the inverse function for the function For A

{3b > 0): g h'(PMb\n)) = (lg)

„ An+i — \n

n=0

there exist a function F G Dl(A, a set E C [0, and a constant ft > 0 such, that inequalities (7) hold for all x G E and h — meas (E) =

n-2

Proof. We denote k1 = k2 = 1, nn = rk, (n * 3), where

k=1

ri = max Ibtpi(bX2), ^ ^ ^

rk = ma^ibpi(bXk+i) - bpi(bXk), --1—— ) ( k ^ 2),

^ Xk+i — Xk J

u+1 — Ak and we also choose

n

00 = 1, an = exp j — ^ Kk(Xk — Xk-1)} (n * 1).

k=1

We prove that the function F defined by series (4) with the above defined coefficients (an) and the exponents ( Ara) belongs to the class ^1(A, $1). Since the condition

EX - X <

n Xn+1 — Xn n=0

ln n

implies n2 = o(Xn) (n ^ we have —--> 0 (n ^ By the construction,

Xn

ln an-1 — ln an Kn = -^^T- (n * 1)

ln a

and Kn ^ + w (n ^ Therefore Stolz theorem yields that--^ (n ^ and

Xn

by Valiron formula [9] the abscissa of the absolute convergence of series (4) is equal to i.e., F e D(A).

Moreover, it is known that in the case nn ^ (n ^ we have

Vx e [xn, Kn+1) : ^(x,F) = anexK, v(x,F) = n. (19)

Since by the construction

n-2 1

Kn ^ bifi1(bXn-1) + y] --r- ^ 2bIfi1 (bXn-1) (n > U0),

t~1 Ak+1 — Ak

for sufficiently large n for all x e [nn, xn+1) we have

2x

lny(2x, F) = ln y(x, F) + / Xv(t)dt * xXu(x)

* * 1 $>(Kf) * I$1 (!)-

Hence, for x * x0 we have

1 / x \ \nß(x,F) £ -x*i{Tb)

and thus F e Di(A, $i). We observe that

1

Kn+i - Kn = T'n-i ^ X _ X- (n ^ 1).

X n - Xn i

For x G

Kni Kn + i

^n—^n. — 1

we have

a i gx1 et i Gx1

-^7-= n- x\-= exp{(Xn - Xn-i)(Kn - x)} ^ := ß, (20)

ß(x,F ) anexÄn

n=1

and, therefore, for x G E = (J xn, xn + — , by choosing n = u(x, F) we get

( a 11 \

F(x) ^ an-iexXn-1 + anexX" = ^(x, F) 1 + n-1 xX ^ (1 + ft)fi(x, F).

v anexA" y

Hence, inequalities (7) are true.

Now we prove that h — meas (E) = By the construction of (xra) for all n ^ 1 we have

K. ^ btpi(b\n-i). (21)

Taking into consideration the Lagrange theorem, the condition h G L+ and inequality (21), we obtain

1

„ , 1 X

h — meas (E) = V^ dh(x) = I h(xra + ----) — h(xn) )

t=1 J t=1\ Xn — Xn-1 /

ra=1 ra=1

Xn

> h'(xra) > h'(6^i(6Ara-i)) _ +

>^A-A 1 > ^ ^ - A 1 _

n=1 n=1

The proof is complete. □

The next criterion is implied immediately by Theorems 6 and 7.

Theorem 8. Let $1 G L, h G L+ and <p1 be the inverse function for the function For each entire function F G D1(A, $1) relation (5) holds as x ^ outside some set E of a finite h-measure uniformly in y G R if and only if (17) is true.

It is worth noting that if condition (16) of Theorem 5 is not fulfilled, that is

i / b \

(3b 1 > 0) : V --- h' Mb A) + T—^ ) _

An+1 — An V An+1 — An y

then for b _ max{61; 2} we have

h'(bipo(bAn))

> —:-7— _

An+1 — An

n=0

Therefore, condition (18) holds and according Theorem 7, there exist a function F G D1(A, $0) a set E c [0, +rc>) and a constant ft > 0 such that inequalities (7) hold for all x G E and

h — meas (E) _

Since for $0(x) _ xa, a > 0, we have D0(A, $0) _ D1(A, $0), from Theorem 5 and 7 we obtain the following theorem.

Theorem 9. Let $0(x) _ xa (a > 0), h G L+. For each entire function F G D0(A, $0) relation (5) holds as x ^ outside some set E of a finite h-measure uniformly in y G R if and only if

i / h \ (Vb > 0) : £ A--h' b(An)1/a + A ° A <

n=0 An+1 — An \ An+1 — An y

is true.

3. ^-measure with a non-increasing density

We note that for each differentiable function h : R+ ^ R+ with a bounded derivative h'(x) ^ c < (x > 0) we have

/ dh(x) = h'(x)dx ^ c / dx.

JE JE <JE

Hence, the finiteness of Lebesgue measure of a set E c R+ implies h — meas (E) < Therefore, according Theorem A, condition (6) provides that the exceptional set E is of a finite h-measure. However, we conjecture that for h E L- in the subclass

DV(A) = {F E D(A) : (3no)(Vn > n,)[M ^ exp{ —Aray(Ara)}]}, y E L,

condition (6) can be weakened significantly. The following conjecture seems to be true.

Conjecture 1. Let y E L, h E L-. If

h'(p(X,n))

n=0 Xn+i - Xn

< +oo,

then for all F e DV(A) relation (5) is true as x ^ outside some set E of finite h-measu,re uniformly in ye R.

h—

The important corollaries for entire functions represented by a laeunarv power series of the form (1) are implied by the proven theorems.

For an entire function f of the form (1) we let F(z) = f(ez), z e C, We observe that as x = lnr, y = <p,

F (x + iy) = F (lnr + iip) = f(r eiip)

and M(x,F) = Mf (r), m(x,F) = mf (r), ^(x,F) = ^f(r), u(x,F) = Uf(r), In addition, for

def

E2 = (re R : lnr e E1} and h1 such th at h[(x) = h'(ex) we have

def f dh(r) f dh(ex) f h — log — meas(E2) = -= --— = dh1(x) = h1 — meas(E1).

Je2 r JE i eX J Ei

The next corollary is implied by Theorem B,

Corollary 1. For each sequence (nk) such, that condition (6) holds and for each function h e L+ there exist an entire function f of the form (1), a constant [3 > 0 and a set E2 of an infinite h-log-measure, i.e.( §E = such that

(Vr e E2) : Mf (r) * (1+[)/if (r), Mf (r) * (1+[)mf (r). (22)

By Theorem 4 we obtain the following corollary.

Corollary 2. Let $ e L, h e L+ and p be the inverse function for the function $. If for an

ln^f(r) * lnr$(lnr) (r * r0) (23)

and

(Vb > 0): V-h'( exp{p(nk) +-}) < (24)

nk+1 — nk V I nk+1 — nk)J

then the relation

f(r eiip) = (1 + 0(1))^ {r)rnvfw ewnvf w (25)

holds as r ^ outside some set E2 of finite h-log-measure uniformly in p e [0, 2ir}.

In fact, it follows from condition (23) that F G D(A, $) with A _ (nk) and it remains to

h1

Denote by E the class of entire functions of positive lower order, i.e.

Af :_ lim lnlnMf (r)/lnr > 0.

T—^ +

By Theorem 8 we obtain the following corollary.

Corollary 3. Let h G L+. In order the relations (3) hold for each function f G E of the form (1) as r ^ outside a set of a finite h-log-measure, it is necessary and sufficient to have

(Vb > 0) : V-h'((nk)b) <

t=0 nk+1 — nk

Acknowledgements

We are grateful to Prof, I.E. Chyzhvkov and Dr. A.O. Kurvliak for helpful comments and corrections in the previous versions of this paper.

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Tetyana Mykhailivna Salo,

Institute of Applied Mathematics and Fundamental Sciences,

National University "Lvivs'ka Polytehnika",

Stepan Bandera str. 12,

79013, Lviv, Ukraine

E-mail: [email protected]

Oleh Bohdanovych Skaskiv,

Department of Mechanics and Mathematics,

Ivan Franko National University of L'viv,

Universytetska str. 1,

79000, Lviv, Ukraine

E-mail: [email protected]