ANALYTIC FUNCTIONS OF INFINITE ORDER IN HALF-PLANE Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 11 (29), No 2, 2022, pp. 59-71 59

DOI: 10.15393/j3.art.2022.11010

UDC 517.537

K. G. MALYUTIN, M. V. KABANKO, T. V. SHEVTSOVA

ANALYTIC FUNCTIONS OF INFINITE ORDER IN

HALF-PLANE

Abstract. J.B. Meles (1979) considered entire functions with zeros restricted to a finite number of rays. In particular, it was proved that if f is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, we prove a similar result for a class of functions analytic in the upper half-plane. The analytic function f in C+ = {z : Im 2 > 0} is called proper analytic if limsupln |/(z)| ^ 0

for all real numbers t e R. The class of the proper analytic functions is denoted by J A. The full measure of a function f e J A is a positive measure, which justifies the term "proper analytic function". In this paper, we prove that if a function f is the proper analytic function in the half-plane C+ of infinite order with zeros restricted to a finite number of rays L^ through the origin, then its lower order equals infinity.

Key words: half-plane, proper analytic function, infinite order,

lower order, Fourier coefficients, full measure

2020 Mathematical Subject Classification: 30D35

1. Introduction, Definitions and Notations. In the 60s of the last century, several authors (L. A. Rubel and B. A. Taylor [16], J. B. Miles [11], J. B. Miles and D. F. Shea [13], [14], and others) started to use the Fourier series method on a large scale for studying of the properties of entire and meromorphic functions. This method is efficient in solution of several general problems of the theory of meromorphic functions and establishes its connections with Fourier series theory. One advantage of this method is its suitability for investigation of functions of fairly irregular growth at infinity and functions of infinite order. In the 80s of the last century, important results in this direction were obtained by A. A. Kondratyuk [5], [6], [7], who generalized the Levin-Pfluger theory of entire functions of

© Petrozavodsk State University, 2022

completely regular growth to meromorphic functions of the gamma-type. At the beginning of this century, the first author of this paper extended the method of Fourier series to delta-subharmonic functions in the halfplane [10].

For meromorphic functions f (z) in complex plane C, the characteristic functions are defined as

m(r,f) = ^/ ln+ If (re*)| dtp,

0

r

N (r, f ) = J n(tJ) — n(0J) dt + n(0, f) ln r ,

0

where n(r, f) is the number of poles of f with regard to their multiplicity in the disk B(0, r) (C(a,r) = {z : |a — z| < r}, B(a,r) = C(a,r), G means the closure of a set G), x+ = max{0; x},

T(r,f) = m(rj) + N(r,f).

The order and the lower order of a meromorphic function f are, respectively, the values

P[/] = limsup ] = i1mmfln^ .

r^TO in r r^TO in r

The order and the lower order of the entire function f are, respectively, the values

H[/] = ilmsup inin,M^), „[/] = iimlnf inln^Z) ,

r^TO in r r^TO in r

where M(r,f)= max If(re%d)I.

In the work [12], J. B. Meles considered entire functions with zeros restricted to a finite number of rays. In particular, it was proved that if f is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity.

We prove a similar result for functions that are analytic in the halfplane. To prove this statement, we use the Fourier series method developed by the first author [10]. Passing to the half-plane is complicated by complex behaviour of the function near the boundary. The difference from

the plane is obvious just in obtaining tests for a meromorphic function to belong to a fixed class (see [9, Teorems 1, 2, and the Example after Theorem 2]).

A special case, when zeros are restricted to the imaginary axis, was considered by the third author [17]. Note that the case of zeros at an arbitrary system of rays is more complicated than one ray. In particular, the proof in this case essentially uses Lemma 2. The main result is the following theorem.

Theorem 1. Suppose f is the proper analytic function in the halfplane C+ of infinite order with zeros restricted to a finite number of rays Lfc through the origin:

Lfc = {z : arg z = ei&k, 0 <9k <n,k e 1,N0, N0 e N} .

Then its lower order equals infinity.

2. Classes of Functions in C+. Let C+ = {z : Imz > 0} be

the upper half-plane. We denote by Q+ the intersection of a set Q with the half-plane C+ : Q + = Q n C+. If 0 < r1 < r2, then D+(r1,r2) = = C+(0,r2)\C+(0,r1) means the closed half-ring.

Let AK be the class of analytic functions f (z) in C+, such that ln |/(z)| possess a positive harmonic majorant in each bounded subdomain of C+. Functions f (z) in AK have the following properties [1]:

a) ln If (z)l has non-tangential limits ln |/(i)| almost everywhere on the real axis and ln If (i)| e L11oc(-w, w);

b) there exists a measure (charge) of variable sign v on the real axis, such that

b

f 1 1 ton y ln If (t + iy)I dt = u ([a, b]) - 2 u ({a}) - - ^ ({b}).

a

The measure v is called the boundary measure of f;

c) du (t) = ln If (i)| dt+da(t), where a is a singular measure with respect to the Lebesgue measure.

For a function f e AK, we define, following [1], the full measure Xf := X by the formula

X(G) = 2n J Im(d^(() - v(G),

C+nG

where ^ is the Riesz measure of in If (z)|.

The full measure determines a function f e AK in the same way as the Riesz measure ^ determines a subharmonic function in C. More precisely, if fi,f2 e AK have the same full measure A, then there exists a real entire function g, such that If1(z)I — in If2(z)I = Im g(z), z e C+.

A function f is called proper analytic in C+ if f is an analytic function in C+ and ilmsupin |/(z)| ^ 0 for each t e R. The full measure of

the proper analytic function is a positive measure, which explains the term «proper analytic function». Note that a proper analytic function is necessarily in the class AK. By J A we denote the class of proper analytic functions in C+.

A function f is called proper meromorphic function in C+ if f can be represented as a quotient of two proper analytic functions. The class of proper meromorphic functions in C+ is denoted by JM.

Proposition 1. [1] JM = AK/AK.

Let f1,f2 e J A. The functions f1 and f2 have no common zeros in C+ if full measures of f1 and f2 are mutually singular.

Proposition 2. [1] For f e JM, the identity f = f1/f2 holds; here f1, f2 are proper analytic functions without common zeros in C+. They are defined uniquely up to a multiplier of the form etfl(z), where g is a real entire function.

Note, however, that the uniqueness of the numerator and denominator in Proposition 2 is not so simple to prove. For example, both could be also multiplied by constant less then 1.

3. Nevanlinna's Characteristic Functions for the Complex Half—Plane. In 1925, R. Nevanlinna [15] (see also [3]) introduced for meromorphic function f (z) in the closed upper half-plane C+ the characteristic functions

r

A(r,f) = 1/ — 1) (in+ If (t)I +in+ If (—t)I) dt, 0

■k

B (r,f ) = — [ in+ If (re^)| sln ^ nr J 0

r

C (rj ) = 2 J + 1) c(tj) dt, 0

where c(r,f) = Y1 sin Vn, rnelPn are poles of f with regard to their multiplicity,

5(r,f) = A(rJ) + B(r,f) + C(r,f).

The following question arises: what is a good analogue of the functions m(r,f) and N(r,f) for C+? R. Nevanlinna [15], A. A. Goldberg [2], A. A. Goldberg and I.V. Ostrovskii [3] reasoned that the analogue of the function m(r, f) is A(r, f) + B(r, f) and the analogue of N(r, f) is C(r, f).

We will exploit other definitions as well to emphasize the connection with the case of the complex plane. These characteristic functions are defined in the widest space where they have meaning. Here we use terminology and definitions from [1].

Let f E JM and let A be its full measure. Let A = A+ — A- be the Jordan decomposition of the measure A. We denote

TT

If -K ^(t, f) '■= ~ ln+ If (retp)l sin= — B(r, f) ,

r J 2 0

r

N(r,f,ro) ■= j ^^ dt = 2(A(r,f) + C(r,f) + 0(1)),

ro

'K

T(r, f, ro) ■= m(r, f) + N(r, f, ro) + m(ro, 1/f) = -S(r, f) + 0(1) ,

where r0 is an arbitrary fixed positive number (one may as well take r0 = 1), which will be omitted in designations provided that this does not cause any misunderstanding; for example, instead of T(r,f,r0) we will write T(r,f), and so on, r ^ r0, X-(t) = X-(B(0,t)).

Definition 1. The order and the lower order of the function f E JM are the values

0 [f ] = lim sup ln(rf(rJ')), a[f ] = liminfln(^-^ . r^oo ln r ln r

4. Preliminaries. Let A be the full measure of a function f E JM. The function f has the following representation in the half-disc C+(0, R) [4]:

M' = - ¿J/^dA(< )+

C+(0,_R) k

+ 2t I dG{Zd^V) In \f (Reilf)\ dtp, z e C+(0,R), (1)

0

dG

where G(z, () is the Green function of the half-disc, -7— is its derivative in

or

G(z,()

the inward-normal direction, and the kernel-'-—, where ( e G+(0, R),

Im(

is extended by continuity to the real axis for \t| ^ R. For the measure A, let

sin k-1

d\k(teiV) = dX(reiV), Xk(r) = Xk(C(0,r)), ke N,

where the function -i-— is defined for — = 0 and — = n by continuity.

sin —

The next relation is Carleman's formula in Grishin's notation [4]:

TT r

1/in I f(re*)I smk —d— = J dt+

0 ro

TT

+ f in If(roeiip)I slnk—d—, ke N.

0

In particular, for k = 1 we have

k r k

1 J In \ f(rei<p)\ sin dip = J ^ dt+ 1 J !n \/(Toeiip)\ sin ^d^p. (2)

0 ro 0

In this notation, Carleman's formula (2) can be written as follows:

T(r, f) = T(r, 1/f ). (3)

The Fourier coefficients of a function f e JM are defined by the formula [10]:

k

Ck ( r, f) = 2/ In \ f(r eiip)\ sink e de, ke N. 0

Let A be the full measure of f E JM; then [10]:

r

Cfc(r, f) = akrk + k E N , (4)

o

where ak = r-kck(r0, f), and from (4) we obtain:

* ^^^ + ¿?JJ_

C+(0,r o)

+ S // — ¿J/^dm , (5)

D+(ro, r) C+(0, r)

where ( = reiP.

From definition of ck (r, f), the inequality follows:

T

2 k

I ck (r, f)I ^ 2k J I ln I f(re*P)II sin p dp, kE N. 0

From this and from equality (3), we obtain

rm( r, f) > ^ I Ck (r, f)I, k E N . (6)

4k

Indeed, (6) follows from the relations

T

I Ck (r, f)I ^ -rj (ln+ I f(r ei P)I + ln+ I1/ f (r e ip)I) sin <pd<p ^ 0

r, f)+m( r, 1/f) ^ 2m( r, f), kE N.

Now we need a lemma:

Lemma 1. If g E JM and Xg = 0, then ln Ig(z)I = ImF(z), where F(z) is an entire real function.

Proof. Remind [8] that the entire function F(z) is a real function if F (R) C R.

Since the full measure of the function g equals zero, then from (1) it follows that for any R > 0

IT

In lg(z)l = R^j dG{ZdRelV) ln lgzeC+(0,R). 0

The right-hand part is a harmonic function in a half-disc C+(0,R), which is extended continuously by zero to the interval (—R, R). As R is an arbitrary positive number, then the function ln lg(z)| is harmonic in the half-plane C+ and is extended continuously by zero to the real axis. By the principle of symmetry, this function is extended as harmonic on the bottom half-plane.

Then there exists a harmonic function h(z) on the complex plane vanishing on the real axis and such that ln lg(z)| = h( z) for Imz > 0.

Let (—h1(z)) be a function harmoniously conjugated to the function h(z). Then F(z) = h1(z) + ih(z) is an entire function, real on the real axis and ln lg(z)| = ImF(z). □

The following lemma [12, Lemma 1.1] is used in the proof of Theorem 1.

Lemma 2. Suppose d1, 62,..., are distinct elements of [0, 2 n). For a real x, let x* denote the unique number in [-n, n), congruent to x modulo 2n. There exists an increasing sequence I = {n{\ of positive integers, such that I has positive density and

(n'e>>* 6 (- 6 ) (7)

for j E 1,N0 and n E F

3. Proof of the Theorem 1. Since Xf restricted to the finite system of rays, then, by formula (5), we obtain for the Fourier coefficients of the function f:

Cn(r, f) = anr- + jr / tn-idX(t)+

f-f nnV0 sin ek J k-i 0

+ * r^rn^n) r dX+) _ * _JMML [r-idX(t),n E N. ^ nn sin Ok J tn+i ^ rnnn sin 9k J

k=i k r0 k=i k 0

Then we obtain

c„(r, ,f) = a„r" + jr "i t"-ldm+

k=1

N

+E

k=1

—ur0n sin 9k J 0

"

sin( QkU) f 1 nn sin 6k J t

( Ï )' - ( ; )

dX(t),n E N . (8)

Integrating by parts twice in (8), we get

^n(f, f) = anrn + £ Sin^ Jtn-1dX(t) +

k=l

2 A sin(0kn) ( n f N(t) , \

- 7 . ! N(r) + rn dt +

n Sin Ok V J tn-1 )

k=1 "o

nnr-Q sin Bk

0

n - 1 N sin( k n) 1

+

sin si

n z—' sin Bk k=1 k

( Ï )' -( ;)

N(t) dt, n E N , (9)

where N(r)

m t2

dt.

f n

By Lemma 2, choose the sequence 1= {nl} such that ( nßj)* E (—— ,—j

N

N

n N

Then y^ sin(9kni) = Y^sin(9kni)* > N0 sin — = N. Note that

k=1

k=1

tr" s2n(°kni! It"'-1*iMt) > 0,ni E N.

=1 —Uir2 1 sin 9k

k=1

From (9) with n = nl, I E N, we obtain

I ^ (r,f )| > n^NIÎ!:) + f N(t)

t"1 > n \ rnl tni+1

dt - Iani I,ni E N .

:iq)

n

n

If the function N(r) has infinite order, then the integral in the right-hand part of the last inequality is unlimited as r ^ x>, because

[Nit) N(r)

—\rdt ^ ^^, n e N, J tn+1 nrn

r

and right-hand part of this inequality can be made arbitrarily large by suitable choice of r. By this inequality, inequality (6), and (10), we receive the demanded statement.

If N(r) has a finite order, then there exist positive numbers K > 0 and p > 0, such that N(r) ^ Krp for all r > 0. It is possible to consider non-integer p. From here it follows that

2 r 2 r

K2prp > N(2r) M dt> X(r)J % = ^ ,

r r

i.e.,

A(r) ^ K2P+1 rp+l.

In this case, from the paper [10, Theorem 3] it follows that there exists the function gl e J A of order p and with full measure A. Then the function g = f /g 1 e J A and Aa = 0.

According to the Lemma 1, lg(z)| = exp(ImF(z)), where F(z) is an entire real function

F (*) = E

'n n=0

an,Z

The fact that an e R, for all n e N, is proved by termwise derivation of the Taylor series of the function F( ) at the point = 0.

If the only finite number n = 0, then F( ) is a polynomial, hence and f have a finite order: this contradicts the condition. Further,

ck (r, g) = akrk, k e N . From this and (6), we obtain

k ^|ak | rk

rm(r, g) ^ —1 Ck(r\ g)1 -

4k1 kV 4k

The last inequality is valid for any fixed natural k. It follows that the function ( ) has infinite lower order.

Further, from the elementary inequality (ln(ab))+ ^ (lna)+ — (ln&)+, which is valid for b > 1, we obtain

m(r, 9) ^ m(r, f) + m(r, 1/91) < m(r, f) + T(r, g\).

Then, from the inequality

liminf rm(r, g) ^ liminf rm(r, f) + limsup rT(r, g\)

r^rx r^rx r—t-oo

and the relations

lim inf rm(r, g) = <x, lim sup rT(r, g-\) ^ p < ,

it follows that

liminf rm(r, f) = m .

The theorem 1 is proved. □

Remark. In Theorem 1 the boundary measure v of f, generally speaking, is not zero. It is a non-negative measure.

The boundary measure of a function f e AK can be sign-variable, its full measure can also be sign-variable. So, I think it has merely been observed that the proof of Theorem 1 does not apply to functions in the class AK. The question whether there exists a specific f in AK of infinite order and finite lower order i still open.

Acknowledgment. We gratefully thank the referee for careful reading of the paper and for the suggestions that have greatly improved the paper.

The work is supported by the Russian Science Foundation (project No 22-21-00012, https://rscf.ru/project/22-21-00012/).

References

[1] Fedorov M. A. and Grishin A. F. Some Questions of Nevanlinna Theory for the Complex Half-Plane. Math. Physics, Analysis and Geometry (Kluwer Acad. Publish.), 1998, vol. 1, no. 3, pp. 223-271.

[2] Goldberg A. A. Nevanlinna's lemma on the logarithmic derivative of mero-morphic function. Math. Notes, 1975, vol. 17, no. 4, pp. 310-312.

[3] Goldberg A. A. and Ostrovskii I. V. Distribution of Values of Meromorphic Functions. English revised edition Amer. Math. Soc.: Providence, RI. 2008.

[4] Grishin A. F. Continuity and asymptotical continuity of subharmonic functions. Math. Phys. Anal. Geom., 1994, vol. 1, no. 2, pp. 193-215. (in Russian)

[5] Kondratyuk A. A. The Fourier series method for entire and meromorphic functions of completely regular growth. Math. USSR-Sb., 1979, vol. 35, no. 1, pp. 63-84.

[6] Kondratyuk A. A. The Fourier series method for entire and meromorphic functions of completely regular growth. II. Math. USSR-Sb., 1982, vol. 41, no. 1, pp. 101-113.

[7] Kondratyuk A. A. The Fourier series method for entire and meromorphic functions of completely regular growth. III. Math. USSR-Sb., 1984, vol. 48, no. 2, pp. 327-338.

[8] Levin B. Ya. Distribution of Zeros of Entire Functions. English revised edition Amer. Math. Soc.: Providence, RI. 1980.

[9] Malyutin K. G. Fourier series and 5-subharmonic functions of finite 7-type in a haff-plane. Sb. Math., 2001, vol. 192, no. 6, pp. 843-861.

DOI: http://doi.org/10.1070/SM2001v192n06ABEH000572

[10] Malyutin K. G. and Sadik N. Representation of subharmonic functions in a haff-plane. Sb. Math., 2007, vol. 198, no. 12, pp. 1747-1761.

DOI: http://doi.org/0.1070/SM2007v198n12ABEH003904

[11] Miles J. B. Quotient representation of meromorphic functions. J. d'Analyse Math., 1972, vol. 25, pp. 371-388.

[12] Miles J. B. On entire functions of infinite order with radially distributed zeros. Pacif. J. Math., 1979, vol. 81, no. 1, pp. 131-157.

[13] Miles J. B. and Shea D. F. An extremal problem in value distribution theory. Quart. J. Math. Oxford, 1973, vol. 24, no. 1, pp. 377-383.

[14] Miles J. B. and Shea D. F. On the grows of meromorphic functions having at least one deficient value. Duke Math. J., 1976, vol. 43, no. 1, pp. 171186.

[15] Nevanlinna R. Uber die Eigenschaften meromorpher Functionen in einem Winkelraum. Acta Soc. Sci. Fenn., 1925, vol. 50, no. 12, pp. 1-45.

[16] Rubel L. A, Taylor B. A. Fourier series method for meromorphic and entire functions. Bull. Soc. Math. France., 1968, vol. 96, pp. 53-96.

[17] Shevtsova T. V. Analytic functions of infinite order on the half-plane with zeros on imaginary axis. Vestn. Bashkir. Gos. Univ. Ser. Mat., Mechanics, 2021, vol. 26, no. 2, pp. 287-292. (in Russian)

DOI: http://doi.org/10.33184/bulletin-bsu-2021.2.3

Received November 10, 2021. In revised form, May 3, 2022. Accepted May 4, 2022. Published online May 13, 2022.

K. G. Malyutin Kursk State University 33 Radischeva str., Kursk 305000, Russia E-mail: malyutinkg@gmail.com

M. V. Kabanko Kursk State University 33 Radischeva str., Kursk 305000, Russia E-mail: kabankom@mail.ru

T. V. Shevtsova Southwest State University

50 Let Oktyabrya Street, 94, Kursk 305040, Russia E-mail: dec-ivt-zao@mail.ru