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.

REFERENCES

1. P.C. Fenton. The minimum modulus of gap power series // Proc. Edinburgh Math. Soc. II. 21:1, 49-54 (1978).

2. P. Erdos, A.J. Macintyre. Integral functions with gap power series // Proc. Edinburgh Math. Soc. II. 10:2, 62-70 (1954).

3. O.B. Skaskiv. Maximum of the modulus and maximal term, of an integral Dirichlet series // Dop. Akad. Nauk Ukr. SSR, Ser. A. 11, 22-24 (1984). (in Ukrainian)

4. R.P. Srivastava. On the entire functions and their derivatives represented by Dirichlet series // Ganita. 9:2, 82-92 (1958).

5. O.B. Skaskiv, M.N. Sheremeta. On the asymptotic behavior of entire Dirichlet series // Matem. Sborn. 131(173):3(11), 385-402 (1986). [Math. USSR. Sb. 59:2, 379-396 (1988).

6. T.M. Salo, O.B. Skaskiv, O.M. Trakalo. On the best possible description of exceptional set in Wiman-Valiron theory for entire functions // Matem. Stud. 16:2, 131-140 (2001).

7. T. Salo, O. Skaskiv. On the maximum modulus and maximal term absolute convergent Dirichlet series // Mat. Visn. Nauk. Tov. Im. Shevchenka. 4, 564-574 (2007). (in Ukrainian).

8. M.R. Lutsyshyn, O.B. Skaskiv. Asymptotic properties of a multiple Dirichlet series // Matem. Stud. 3, 41-48 (1994). (in Ukrainian).

9. S.Mandelbrojt. Dirichlet series. Principles and methods. Reidel Publishing Company, Dordrecht (1972).

Tetyana Mykhailivna Salo,

Institute of Applied Mathematics and Fundamental Sciences,

National University "Lvivs'ka Polytehnika",

Stepan Bandera str. 12,

79013, Lviv, Ukraine

E-mail: tetyan.salo@gmail.com

Oleh Bohdanovych Skaskiv,

Department of Mechanics and Mathematics,

Ivan Franko National University of L'viv,

Universytetska str. 1,

79000, Lviv, Ukraine

E-mail: olskask@gmail.com