CC BY
6
УДК 517.53
Coefficient Multipliers for the Privalov class in a Disk
Eugenia G. Rodikova*
Bryansk State University Bezhitskaya, 14, Bryansk, 241036 Russia
Received 02.05.2018, received in revised form 20.07.2018, accepted 06.09.2018 We obtain exact estimates of the growth and the Taylor coefficients of analytic functions from the Privalov classes in the unit disk. Also we describe coefficient multipliers from the Privalov classes into the Hardy classes.
Keywords: Privalov classes, Taylor coefficients, coefficient multiplier, maximal growth, analytic functions.
DOI: 10.17516/1997-1397-2017-11-6-723-732.
Introduction
Let C be the complex plane, D be the unit disk on C, H(D) be the set of all functions,
holomorphic in D. For all 0 < q < we define the Privalov class of function ng as follows
(see [8]): ^
ng = ( f € H(D) : sup ^ T (ln+ \f (reie)) dd < .
{ 0<r<i 2n J)
Note that the classes ng were first considered by 1.1. Privalov in [8]. In the case 1 < q < the Privalov spaces were studied by M. Stoll, V. I. Gavrilov, A.V. Subbotin, D.A.Efimov, R. Mestrovic, Z. Pavicevic, etc. The monograph [2] contains a brief overview of their results. The case 0 < q < 1 has been little studied in the scientific literature. In this paper we obtain exact estimates for the maximum modulus and the Taylor coefficients of functions from the classes ng (0 < q < 1) (Section 1), on this basis we describe coefficient multipliers from the Privalov classes ng (0 < q < 1) into the Hardy classes Hp (0 < p < (Section 2).
Notice that the problem of describing the Taylor coefficients for analytic functions of the Nevanlinna class was first solved by S. N. Mergelyan in the early 20th century (see [9, p. 152]). Later on these questions in the Hardy classes were investigated by G.Hardy and D.Littlewood, A.A.Friedman (see [3]), in V.I. Smirnov's classes by N. Yanagihara [20], in the Privalov classes ng (q > 1) by M. Stoll [18], in the plane Nevanlinna classes by S. V. Shvedenko [17], in the weighted classes of analytic functions in a disk with restrictions on the Nevanlinna characteristic by F. A. Shamoyan and E. N. Shubabko [14], and by the author of this paper (see [10]).
As the authors observe in [2, p. 148], the notion of a coefficient multiplier arises naturally in the study of asymptotic properties of the Taylor coefficients for functions from certain classes. In a simplified form, the problem is posed as follows: which factors the Taylor coefficients of a function of a given class must be multiplied by for them to acquire special properties, for example, being bounded or form an absolutely convergent series. Requiring that the resulting
* [email protected] © Siberian Federal University. All rights reserved
products be Taylor coefficients of functions from some other class, we arrive to definition of the coefficient multiplier.
Let X and Y be some classes of functions analytic in the unit disk D.
Definition 1. The sequence of complex numbers A = {Akis called a coefficient multiplier
from class X into class Y if for any function f £ X, f (z) = ^ akzk, we have A(f )(z) =
k=0
= £ Akakzk £ Y. It is denoted by CM(X,Y).
k=0
Numerous works are devoted to the description of multipliers in various classes of holomorphic functions. We recall some of them: [1,2,4-6,11,13-16,19], and etc.
1. On maximal growth and the Taylor coefficients for functions from the Privalov classes
The following statement is valid: Theorem 1.1. If f £ nq then
ln+ M(r, f) = o((1 - r)-1/q), r ^ 1 - 0, (1)
where M(r, f) = max \f (z)\.
\z\=r
Remark. Throughout the paper we study the Privalov class nq with the parameter 0 < q < 1. Also, unless otherwise stated, we denote by c, c1,... ,cn(a, 3,...) arbitrary positive constants depending on a, 3,..., whose specific values are immaterial.
Proof. We choose an arbitrary point z0 £ D and by definition put KZo = {Z £ D : \Z — z0\ < ^(1 — \z0\)}. Let dm2 be the planar Lebesgue measure. From the inequality (see [7, p. 144], Theorem 9.1.1, equation(9.3)) which holds for all 0 < q < 1:
(ln+ \f(z0)\)q < —Cqzo{)2 f (ln+ \f(Z)Dqdm2((),
Kzn
we obtain
(ln+ \f(zo)\)q < (1 __Cq]zoj f (ln+ \f(pe16)l)qdedp,
\zo\--n
whence we have:
(1 _ \Zo|) \zo\- —01 <P<\zo\ + J-n
(ln+ \f(zo)\)q < _ . .sup _ 1 1 I (ln+ \f(pei6)\)qde <
^ Cq sup i ( ln+ \f (pei6 )\)q de.
0<P<1J-n
(1 — \z0\) 0<p<1.
Now the required estimate (1) follows. □
Theorem 1.2. If J2 akzk is the Taylor series expansion for a function f € ng, then
k=0
ln+ \ak\ = o (k ,k ^ (2)
Proof. We prove this theorem using the method of S. N. Mergelyan (see [9, p. 152]). From the Cauchy inequality and the estimate (1) of Theorem 1.1 it follows that for any arbitrarily small e > 0 there exists re € (0,1) such that
\ak \ < r-k exp j e(1 - r)-1 j ,re <r < 1, n = 0,1,. . . , (3)
which is equivalent to
ln+ \ak \ ^ e(1 — r)-q — k ln r, re < r < 1, n = 0,1,. .. . (4)
Introduce the function 1
^(r) = e(1 — r)-q — k ln r.
We investigate it to find its exact lower bound. Calculate the derivative:
e 1 k
= -
■ r ) q 1 r
q (1 — r)<
We find the minimum of the function ^(r), solving the equation (r) = 0:
er
e--= k. (5)
q (1 — r)1+1
The solution of this equation exists and unique on the interval (0, 1). For convenience, we introduce the following notation:
1 1 — rk
Ck = ' Sk = c ,— ,
bsjrl O^jFk
where 5 > 1.
We can assume that sk <ck < 1. Indeed, the inequality sk <ck is obvious. Now, ck < 1 is equivalent to
VTk > 1, (6)
while sk < 1 is equivalent to
v/52T4 — 5
V^k >-2-' (7)
and (6) follows from (7).
In the new notation equation (5) takes the form:
qS2 s
or
1 [Ck\
2 A skj
1+i sk
1
c
kqS2
Since ck < 1, the last equality implies the estimate:
q
f e \ q+1
sk HkqSî) ■ (8)
e
k
e
From the same equation we obtain:
(StY=(
Taking into account the estimate (8), we have:
q (ks\q—2\ 1-q
mq < (?) ^ *.
(9)
Using (8), (9), we estimate the value of the function ¿(r) at the minimum point r = rk :
)= e(1 - rk) q - k ln rk.
The application of (9) yields:
<(rk ) ^ £ ( - ) • k 1 + 9 — k ln rk.
To estimate the last term we note that
(rk)-1 - r£ exp (-1 lnrk) - exp(1 lnrk)
— sh Qln r^ = sh ^—2ln r^j
whence
Thus we have:
— ln rk = 2arc-h—k— < 2 —^, k 2 2 '
—k ln rk ^ k-k—.
sM— 2ln rk) =—2—'
¿(rk) < k £ (qS2) ^q • + . The required estimate (2) follows.
(10) □
2. Description of the coefficient multipliers
from the Privalov classes into the Hardy classes
Theorem 2.1. Let 0 <p < A = [Xk^^ C C. For A = CM(n, Hp) it is necessary and sufficient to have
|Ak| = O ^exp (—c • k ^j) ,k ^ (11)
for some c> 0.
The proof of this theorem is based on auxiliary statements.
Lemma 2.2. (see [2, Lemma 9.7]) Let F and H be linear classes of holomorphic functions in the unit disk D with metrics, convergence in which is not weaker than the uniform convergence on compact subsets of D. Then each coefficient multiplier from class F into class H is a linear and closed operator.
To formulate the next Lemma we introduce a metric on the class ng:
1
P(f, 9) = sup — In« (1 + \f (reie) — g(reie) |) d0.
0<r<1 2n J-n
2
2
Lemma 2.3. The class nq with respect to the metric (12) is an F-space.
Proof. The proof of this statement is equivalent to establishing the properties a)-d) of a metric (see [12]):
a) p(f, g) = p(f - g, 0) is obvious.
b) ng is a complete metric space.
Let {fn} be an arbitrary fundamental (Cauchy) sequence from the class nq, i.e. Ve > 0 3N(e) > 0 : Vn, m > N ^ p(fn, fm) < e. We show that it converges to some function f G nq. Note that functions ln(1 + |fn|) are subharmonic in D. In the same way as in the proof of Theorem 1.1 we use the equation from Theorem 9.1.1 of [7, p. 144] to obtain:
lnq (1 + IfnRe'n - fm(ReiLp)|) Cqm2 sup f lnq (l + lfn(reie) - fm(reie)|) dd,
(1 - R)2 0<r<U-n
2nc
lnq (1 + Ifn(ReiV) - fm(RelLp) |) < p(fn, fm),
whence we have
fnRp - fmReJm ^ 0,n,m ^
for all 0 < R < 1, p G [-n, n]. So the sequence {fn} converges uniformly in the unit disk for some function f G H(D). Now we prove that f G nq.
1 1
sup — (ln+ \f (reie)\)9dd < sup — ln (1 + \f (reie)\)9dd <
<r<i 2n J-n 0<r<1 2n J-n
1 t'n
< sup — ln9 (1 + \f (reie) - fn(reid)\ + \fn(reie)\) dd.
0<r<1 2n J-n
0<r<1 2n J-n 0<r<1
1 I ln9 (1 + \f (reie) f (reie)\ , \f freie
r<1
Since (a + b)9 < (a9 + b9) for any a> 0, b > 0, 0 < q < 1, we have
1 pn
sup— / (ln+ f (reie )|) q dd <
0<r<1 2n J -n
1 n
< sup — (lnq (1 + ^(e) - fn(reie)|) +lnq(1 + me)|))dd < const.
0<r<1 J
We conclude that nq is a complete metric space.
c) If f, fn G nq and p(fn, f) ^ 0, n ^ then for any 3 G C we have p(3fn,3f) ^ 0, n ^ +TO.
For 3 < 1 the property immediately follows. Assume that |3| > 1, without loss of generality we may take 3 > 1. Since the sequence {fn} converges, it is fundamental (Cauchy). As stated above, from that follows its uniform convergence inside D.
Since for any 3 > 1 and x > 0 the estimate (1 + ¡3x) < (1 + x)P is valid, we have
1 [n
p(3fn,3f)= sup — lnq (1 + 3 fn(reie) - f (reie)|)dd < 0<r<1 2n J-n
< sup ^ f lnq (1 + fn(reie) - f(reie)|)dd = 3qp(fn,f),
0<r<1 2n J-n
whence the property c) follows.
d) If 3n, 3 G C and 3n ^ 3, then we have p(3nf, 3f) ^ 0, n ^ for any function f G nq. The property immediately follows from the inequality
ln(1 + 3n - 3 f |) < ln(1 + f |) + ln(1 + 3n - 31). Lemma 2.3 is proved. □
Lemma 2.4. Let the sequence of complex numbers {Aksatisfy the condition:
I k=1
Ak I = O (exp (—ck • k ,k ^ (13)
for an arbitrary positive sequence {ckck ^ 0, k ^ Then there exists c > 0 such that
for all k € N the condition (11) is true.
The proof of Lemma 2.4 repeats the arguments of the article by N.Yanagihara [19] (see Lemma 1) with the exponent -.
q + 1
Lemma 2.5. Let 0 < q < 1,
c
9(z) = exp ---r, z € D, (14)
(1 - z)5
1
where 0 < c < -, Y1 an(c)zn be the Taylor series expansion for g. Then the following estimate
q n=i
is valid:
Ian(c)I > exp (c^ • n. (15)
The proof of Lemma 2.5 repeats the arguments presented in the author's thesis (see [11,
p. 104], Lemma 2.7) with the exponent of -. The method goes back to Mergelyan S. N. (see [9]).
As stated above, from p(fn, f) ^ 0, n ^ it follows that the sequence fn(z) uniformly
+^^ . . . , converges to f (z) in D. Therefore if fn(z) = J2 ak'zk and f (z) = J2 akzk, then a^' ^ ak,
k=0 k=0
n ^
Let X be an F-space consisting of complex sequences {bk}k such that convergence of a sequence 3(n' = {b^} to 3 = {bk}, n ^ implies coordinate-wise convergence ^ bk, n ^ k = 0,1, 2,... .
Consider a coefficient multiplier A = CM(nq,X). By Lemma 2.2, A is a closed operator, therefore by the closed graph theorem (see [12]) A is a continuous operator and it maps bounded sets in the class ng into bounded sets in X.
Proof of Theorem 2.1. Let A = {Ak}+=°i be a coefficient multiplier from the class ng into the class X. We prove that there exists c > 0 such that the estimate (11) is valid, i.e.
|Ak| = O (exp (—c • k, k ^
According to Lemma 2.4, it is enough to show that A satisfies the estimate (13) for some positive infinitesimal sequence {ck}.
We choose the sequence {ck} so that the following estimates are valid:
k-25 < ck < 1. (16)
We consider in ng the sequence of functions
fk (z) = g(rkz) = exp-ckk-r, k = 1, 2,..., 0 < rk < 1, (17)
(1 - rkz) 5
satisfying the conditions of Lemma 2.5, and the double inequality
f^J—ï < rk ^1 — B—1, Bk = ^i^y , Bk > 1' (18)
Ck — 1
where jk is a positive infinitesimal sequence such that — < k q, k = 1, 2,... .
Yk
It is obvious that rk — 1 - 0, k —
Let us show that fk G n9.
r>n /
Ck
sup 2- f (ln+ \fk(reie)\)9dd = sup 2- f (ln+
0<r<1 2n J-n 0<r<1 2n J-n ^
exp -
(1 — rk reie ) q
de c
1 T ck m 1 r ck
< sup — --k—T^T dd = sup — k dd =
0<r<1 2n J-n |1 - rkrei<j | 0<r<1 2n J1 - 2rkr cos d + (rkr)2
(1-rk r n
f ... + f ... I <
0 Ukr
/ 1-rkr n
< sup ck if --1-- dd + / —= 1 dd
0<r<1 n V 0 (1 - rkr) 1-ir
cq l n dd 1 cq ( n n \ = sup 11+/, = | = sup 1 + --- ln -- .
0<r<1 ^ Jkr (1+ rkr)2 (1 )2j 0<r<1 ^ (1+ rkr) 1 - rkrJ
Taking into account the evident inequality ln x < x, Vx > 0, and the condition (18) we conclude sup 2- f (ln+ fk(reie)|)qdd < sup i (1 + (1 f ) = ^ U + 77^] < Yl.
0<r<1 2n J0<r<1 n \ (1 - (rkr)2)J n \ (1 - r2)J k
We prove that {fk} is a bounded sequence in the class nq, i.e. we show that there exists a real number 0 < A < 1 such that for all natural numbers k the inequality p(Afk, 0) < e is valid, where e is a fixed positive number (see [12, p. 31]). First, we prove that
ln(1 + |A||g|) < (ln(1 + |A|)+ln+ |g|). (19)
Indeed, if |g| < 1 then |A||g| < |A|, and the estimate (19) follows immediately. If g > 1 then ln(1 + |A||g|) < ln(|g| + |A||g|) < ln(1 + |A|) + ln+ |g|.
Now we prove the inequality p(Afk, 0) < e. Since (a + b)q < (aq + bq) for any a > 0, b > 0, 0 < q < 1, using the estimate (19) we get
1 Cn
1 I , q /. , , „ ,
1
p(Afk, 0)= sup — lnq (1 + Afk^9 )|) dd < lnq (1 + |A|) + (Yk)q.
0<r<1 2n J-n
Since Yk = o(1), k — Ve > 0 3k0 G N : Vk > k0 the following inequality holds: Yk < d^k.
2
Choosing Ako so that ln(1 + ^^ |) < ^2, we see that starting from some number k0 all elements
of the sequence {fk} are contained in a ball of radius e.
Since nq is an F-space, for all numbers k < k0 there exists a positive number Ak such that VA G C, |A| < Ak the following inequality is valid: p(Afk, 0) < e. Assuming A0 = = min(A1, A2,..., Ako), we obtain that for all |A| < A0 the sequence {fk} is contained in a ball of radius e, i.e. p(Afk, 0) < e.
Owing to arbitrariness of the choice of e, we conclude that {fk} is a bounded sequence in the class nq.
Since the sequence {fk} is bounded in nq, we get that the coefficient multiplier A(fk) is bounded in X.
Let X = Hp. We have
IIA(fk)\\h? < C, C> 0. Fix a number k € N. If fk(z) = £ a{n'zn € Hq, then A(fk)(z) = £ Ana{n'zn € X, therefore
n=0 n=0
(see [3, c. 98])
lAna^I < cp||A(fk)\\hp ■ np-1, for 0 <p< 1, lAna^I < cp||A(fk)\\hp, for 1 < p < to,
whence we have
IAna(k'I < C ■ cp ■ n1-1, for 0 <p< 1, (20)
IAna<k)I < C ■ cp, for 1 < p < to, (21)
where
cp is a positive constant depending on p.
an
p
Since fk (z) = g(rk z), we have oin' = an(ck )rn. According to Lemma 2.5,
Ia^'I > rnk exp (4r n. Using the inequality (18), we have:
2 k
\a[k)\ > — 2 exp (of1 kq+r) . (22)
k-1
From (20), (22) we obtain the following estimate for 0 < p < 1:
2 k
IA k I < C ■ c'p ■ M - -—) 2 ■ kp-1 ■ exp —f1 k 5+r)
and applying now the estimate (16) we have:
IAk I < Cexp (-c5+ ki+r) . (23)
From the inequality (23), applying Lemma 2.4, we conclude that the estimate (11) holds. Analogously, for 1 < p < +to from (21) and (23) we obtain the required estimate.
We prove the converse statement of Theorem 2.1. Let a sequence A = {Ak} satisfy the
+w
condition (11) of Theorem 2.1 and f € nq, f (z) = ^ akzk. From Theorem 1.2 it follows that
k=0
IakI < Ci exp [ekk, £k | 0. c
Choosing a number k0 such that ek < ^ for all k > k0, we obtain:
Ak ak \ < C2 exp 2k .
+^^ / c r \
Since the series J2 exp (—- k5+r 1 converges, A(f)(z) € X for any specified choice of class X.
2
J'
k=0 v 2
Theorem 2.1 is proved. □
Remark. Note that the method of the proof of Theorem 2.1 goes back to N. Yanagihara's work [19].
The immediate consequence of Theorem 2.1 is the statement that the estimates of Theorems 1.2 and 1.1 are unimprovable. The proof of this statement goes in the same manner as in the R. Meshtrovic's article (see [2, p. 152], Consequences 9.24, 9.26).
The author thanks Professor F. A. Shamoyan for carefully reading of the manuscript and helpful comments.
The work was financially supported by Russian Foundation for Fundamental Research, project 18-31-00180
References
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О коэффициентных мультипликаторах классов Привалова в круге
Евгения Г. Родикова
Брянский государственный университет им. ак. И. Г. Петровского
Бежицкая, 14, Брянск, 241036 Россия
В статье получены точные оценки роста и коэффициентов 'разложения в ряд Тейлора функций из классов И. И. Привалова, полностью описаны коэффициентные мультипликаторы из класса Привалова в классы Харди.
Ключевые слова: класс Привалова, коэффициенты Тейлора, коэффициентный мультипликатор, рост, аналитические функции.
