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7также для систем газоанализа и мониторинга атмосферы. Работа выполнена при поддержке национального проекта «Наука и университеты» (проект FSWR-2021-012) за счет субсидии федерального бюджета на финансовое обеспечение государственного задания на выполнение научно-исследовательских работ.
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ОЦЕНКИ КОЭФФИЦИЕНТОВ ТЕЙЛОРА ДЛЯ КЛАССА СПИРАЛЕВИДНЫХ ФУНКЦИЙ ПОСРЕДСТВОМ ДИФФЕРЕНЦИАЛЬНОГО ПОДЧИНЕНИЯ
Султыгов М.Д.
профессор кафедры математического анализа, кандидат физико-математических наук, ФГБОУ ВО «Ингушский государственный университет»
SHARP COEFFICIENT ESTIMATES FOR A CERTAIN GENERAL CLASS OF SPIRALLIKE FUNCTIONS BY MEANS OF DIFFERENTIAL SUBORDINATION
Sultygov M.
Professor of the Department of mathematical analysis, Candidate of physical and mathematical Sciences, Ingush state University, Magas
Аннотация
В настоящей работе получены несколько точных оценок для коэффициентов Тейлора в некотором общем классе S^ (А, В) спиралевидных функций в полных ограниченных кратно круговых областях D с Сп, который определяется с использованием принципа дифференциальной субординации. Результаты статьи дополняют многочисленные точные оценки тейлоровых коэффициентов в различных подклассах изучаемого класса.
Abstract.
In this paper, we obtain several exact estimates for the Taylor coefficients in a certain General class of (A, B) helical functions in complete bounded multiples of circular domains D с Cn, which is determined using the principle of differential subordination. The results of the article complement numerous accurate estimates of Taylor coefficients in various subclasses of the class under study.
Ключевые слова: Коэффициенты Тейлора, спиралевидные функции, полные ограниченные кратно круговые области, принцип подчинения, гипотеза Бибербаха, области Рейнхарта.
Keywords: Taylor coefficients, spiral functions, complete bounded multiples of circular domains, the subordination principle, the Bieberbach hypothesis, Reinhart domains.
Introduction.
In 1916, L. Bieberbach [1] proposed the famous hypothesis: that
|cn| <n,n = 2,3,... Holds for all regular and univalent functions in the unit circle |z| < 1 functions f(z) = z + %n=2 cnzn.. The proof of the hypothesis was obtained only in 1985 by the French mathematician L. de Brange [2].
The purpose of the article is to construct effective exact estimates of Taylor coefficients for a certain General class of helical functions by means of differential subordination in the form of a multidimensional analog of the Bieberbach hypothesis in Reinhart domains. We consider functions that are holomorphic in complete bounded multiples of circular domains D c Cn or in
their subdomains Dr = rD, where D is the closure of the domain D and r E (0,1).
The results of the article complement numerous accurate estimates of Taylor coefficients in various subclasses of the class under study.
Let's call f(z) e H(D c Cn) a function of class Qd [3, 10] if in D c Cn has a decomposition
f(z) = 1 + ^=1 akzk (1) and F(zk) = zkf(vtzk,... , zk,... ,vnzk), as a function of variable zk, is univalent in the section of the domain D c of the complex line
-{zk=^:vmEC\{0},m = 1.....kV. Vm
function F(zk) =
dt
f(z,t) = h(z,t)L1[f(z,t)].
(5)
Pv[k]=\zlf
when vm = 0,
the
zkf (0,..., zk,... ,0)is univalent in section Am = D n {zm = 0:m = l,...,k -l,k + 1,...,n}. Definition 1. Holomorphic function f(z) e H(D c Cn) that satisfies the condition
(2)
Re?!^l>o
№
A <
We will call the A- Spirallike function, -- <
n 2'
Here the operator L1f(z) has the form L1[f(z)] = f(z)+Tl]=1Zjd-^ [4, 10].
In the case of a single complex variable, this class was introduced by Spacek L. [5], and showed that the functions of this class are univalent.
In 1967, R. Libera [6] extended this definition to A - Spirallike functions of order a, 0 < a < 1, of a single complex variable.
Definition 2. We say that f(z) e H(D c Cn) belongs to the class SD (A, a), if and only if
(3)
f(z)
To simplify writing, all the arguments below are performed for the case of two complex variables, which is easily transferred to the case of many complex variables.
Theorem 1 [7, 37]. A sufficient condition for the holomorphic function f(z1,z2),f(0,0) = lto belong to the class SD (A, a) has the form
lk!,k2
|fc|-1
^n
1=0
12(1 - a)e-acosA + j
7+1
EN\[1},N := {1,2,3,...}. (4)
Here |fc| = k1 + k2 and the coefficient estimation (4) is sharp.
Let's recall the principle of differential subordination between two analytical functions. First, we prove the following theorem.
Theorem 2. Let f(z, t) = et + akzk be a holomorphic function in the domain D for any fixed t. Here Ikl ¥ Y'l=1ki,k\ ¥ U1l=1ki\
Then {f(z, t)} forms a chain of subordination if:
1. f(z, t) as a function of t is absolutely continuous, locally uniformly over z E D (in the poly-circular norm).
2. There is a family of -measurable functions h(z, t) E Qd [3,c. 10] for any t, such that for almost all
Proof. Consider a fixed point z E Dro. Let p E (r0,1).Then, by virtue of the completeness of the do-mainD, the point ^ E D if | ^ | <1Q E C1. Consider
the function F(t) = t) as a function of %
and t. Due to the assumptions of the theorem, F ((, t) is holomorphic in measurable, and absolutely continuous in t, locally uniform in £. Due to the assumptions of the theorem, F ((, t) is holomorphic in {, measurable, and absolutely continuous in t, locally uniform inf.
In addition, there is a family of functions
/fz0 \
HQ,t) = h(—,t) satisfying the conditions Re HQ, t) > 0,H(0, t) = 1, such that for almost all t>0
d d giF(f,t) = fH(f.t) — F(f,t)
that is true in virtue of (8) and the fact thatLi[f(i;Z,t)]=^[i;f(i;Z,t)].
Thus, all the requirements of theorem X. Pom-merenke [8] are fulfilled, and we can state that {F(z, t)} forms a chain of subordination, that is, for any
0<t1<t2<ro there is F(z, t1) < F(z, t2). Assuming % = p, passing to the limit at p ^ r0 and using the completeness of the domain Dr, we come to the statement of the theorem.
Using theorem 1, it is easy to prove the criterion for belonging solutions of equation (5) to the class QD [2].This sufficient condition is a generalization of the result of X. Pommerenke (see [8]).
Theorem 3. Let f(z,t) e H(D) be continuously differentiable by t,
0 <t < <x,f(0, t) = et, and satisfy equation (1). If f(z,t) = a0(t)f0(z) + 0(1),fo(0) = 1,
where lim a0(t) = ro, 0(1) - a finite value for a
fixed z E D and t ^ ro,
fo(z) is a non-constant holomorphic function in D, then both the functions f(z, t) and f(z, 0), f0(z) belong to the class QD.
The proof of the theorem consists in reducing it to the case of a function of one complex variable and checking the fulfillment of all the conditions corresponding to the one dimensional theorem X. Pom-merenke.
Definition 3. We will say that the function f(z1,z2) is subordinate to the function g(z1,z2), and write further as f(z1,z2)<g(z1,z2) if f(Dr)c g(Dr)forallr E (0,1).
Definition 4. The Class of generalized helical functions Sp(A,B),
1 < B < A < 1,IAI < - [9, c. 1353,] is the set of all representable f(z1,z2)EH(Dc C2) next to (1) and satisfy the condition of subordination:
LJ(z) 1 + A0(z) ^
ESd(0) [3.C.7]
d
Comment 1. There Are many variants of parameters A and B that could provide interesting subclasses of analytical functions that were previously studied [7, 10-13].
After proving the famous Bieberbach hypothesis [1] (i.e. de Branges theorem [2]) on exact coefficient estimates for functions in the class of one - dimensional functions S, coefficient estimates have also attracted great interest from various other subclasses of the class of normalized analytic functions f(0)= f(0)- 1 = 0 in one-dimensional S and in the class QD in the multidimensional case.
The main purpose of the article is to extend the main results obtained earlier in [7, 10-13].
We give exact estimates of the bounds of coefficients of the studied class of functions, which requires the following Lemma.
Lemma. Let the parameters A, B and p, as well as the integer m bounded --1<B<A<1, |A| <- and |fc| 6 N\{1}, be fixed. Let's also assume that (A - (|fc| - 1)B)2cos2p + (|fc| - 2)2(B2sin2p -1) > 0. (6)
Then 1
(\k\-1)2
(A - B)2cos2ß
\k\-i
+
1\(A
\k\B)2cos2ß
\k\=2
+ (\k\ - 1)2(B2sin2ß-
\k\-2 ,
j-j l(A - B)e-lßcosß
j=0
1)\
-jBl2
(j + l)2
(7)
The proof of the Lemma is based on the principle of mathematical induction for all |fc| 6 W\{1}.
Theorem 4. Let (z) 6 (A, B),-1 < B < A < 1,W<^ , w 6 M{1} and f(z) = 1+Y^k\=iakzk.
Then
\k\-2
lak1,k2l < ^ j=0
l(A- B)e-ißcosß -jBl
J+1
,\k\
E N\{1} ,N = {2,3,...}. (8) The coefficient estimates in (8) are sharp. To see that the coefficient estimates stated by the theorem are accurate, it is sufficient to consider the following function:
1
f(z) =-ï
(1 + Bz)~
-ißcosß
boundary of this region can be represented in the following parametric form: |z1| = r1(т),|z2| = t2(t), 0< t <1, where r1(0) = 0, r1(1) <®,
t'(T) > 0,(0 <T <1) - din r1
and
^CO =
R2exp [- /dinr1(r)] , r2(1) = O.This parametric representation of the D1 domain allows efficient calculation of dkiik2(D1). Indeed, for ^ = k1 + k2 > 0
dkl,k2(D1) = rkl gL) T2k2 , counting 00 = 1.
Note also that if the area fl is a Bicylin-der dzj < R^^^z^ < R2}, then it is obvious that dk k (D) = R«^R22. So, in the case of those regions D whose boundaries are twice continuously dif-ferentiable and analytically convex from the outside, as well as in the case of a Bicylinder, estimates of Taylor coefficients are effective.
Theorem 5. An Analog of the Bieberbach problem for functions f(z1,z2) 66 S^2 (A,B) in a Bi-
2
Rl,R2
cylinder has effective estimates of Taylor coefficients of the form:
lakl,k2(f-.U2Ri,R2)l
\k\-2 „
i n -B) -
<
J=0
j + 1
Theorem 6. Analog of the problem of Bieberbach functions f(z1,z2) E S^ (A,B) in hyperconus Ki = {(z1,z2) E C2. Iz1l + lz2l < 1}, where the boundary of this region is can be represented in parametric form:
dKi = {(zi,z2) E C2.\zi\ = T, IZ2I = 1 — T ,0< t < 1},
'hi
,\k\J V|fc|,
Effective evaluation of Taylor coefficients has the following form:
W^tt-.KDl
N (ki\kl (k2\k2 ^^Q Q
\k\
< If k1b k2 Iii IÎ2
\k\-2
n
1 = 0
l(A- B)e-ißcosß -jBl
J+1
As a final example we give an analog of the Bieberbach hypothesis in a logarithmically convex bounded complete bicircle domain
Dp,q = {(Z1,Z2) 6 c2. + № <1;p =
m .
— ,m,n,q 6 N}.
n J
Note that Dp q 6 (T) if and only if p > 1. In the domain Dp q 6 (T), the parameterization radii r1 (t) and r2 (t) have the form
The Taylor coefficient estimates include the value r1p (t) =
Tq
dkijc2(D) = sup(\zi\kl\z2\k2) for all (zi,z2) e D c C2. For a particular type of domain D, it is important to be able to calculate dkltk2 (D). in order to obtain effective estimates of the Taylor coefficients, the question arises of allocating special classes of domains D for which dkik2(D) can be effectively calculated. Let Di be the domain D whose boundary is twice continuously differentiable and analytically convex from the outside. As proved by A. A. Temlyakov [14], the
1 w —-. jz-\ ''1
Tq + (1- T)p
,ri«(T)=-
(1 - r)q
, rge
Tq + (1 - t)P '
hi hz
dki,k2(f. Dp«) = P Q
00 = 1.
Theorem 7. For functions f(zi, z2) E S§ q (A, B)
in a logarithmically convex bounded complete bicircu-lar domain Dpq, effective estimates of Taylor coefficients have the form:
K^if-: Dp«)1
kl4 + k2P \k\-2 .
(k1q + k2p) 4P T-T fl(A- B)e-lßcosß — jBl
- kl kT~ 11
(k1q) p (kip) 4 1=0
i + 1
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