On a complete description of the class of functions without zeros analytic in a disk and having given orders
For arbitrary $0 ≤ σ ≤ ρ ≤ σ + 1$, we describe the class $A_{σ}^{ρ}$ of functions $g(z)$ analytic in the unit disk $D = \{z : ∣z∣ < 1\}$ and such that $g(z) ≠ 0,\; ρ_T[g] = σ$, and $ρ_M[g] = ρ$, where $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\...
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| Date: | 2007 |
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| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2007
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3360 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509436291842048 |
|---|---|
| author | Chyzhykov., I. E. Чижиков, І. Е. |
| author_facet | Chyzhykov., I. E. Чижиков, І. Е. |
| author_sort | Chyzhykov., I. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:14Z |
| description | For arbitrary $0 ≤ σ ≤ ρ ≤ σ + 1$, we describe the class $A_{σ}^{ρ}$ of functions $g(z)$ analytic in the unit disk $D = \{z : ∣z∣ < 1\}$ and such that $g(z) ≠ 0,\; ρ_T[g] = σ$, and $ρ_M[g] = ρ$, where
$M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$
$ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$. |
| first_indexed | 2026-03-24T02:41:04Z |
| format | Article |
| fulltext |
UDK 517.544+517.547+517.57
I. E. ÇyΩykov (L\viv. nac. un-t)
PRO POVNYJ OPYS KLASU
ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV
IZ ZADANYMY VELYÇYNAMY PORQDKIV
For arbitrary 0 1≤ ≤ ≤ +σ ρ σ , we describe the class Aσ
ρ of functions g z( ) analytic in the unit disk
D = { }:z z < 1 such that g z( ) ≠ 0, ρ σT g[ ] = , ρ ρM g[ ] = , where M r g( , ) = max ( ){ g z :
z r≤ }, T r u( , ) =
1
2
0
2
π
ϕ
π
ϕ∫ +ln ( )g re di , ρM g[ ] = lim sup
ln ln ( , )
ln ( )
r
M r g
r↑
+
− −1
1
, ρT g[ ] =
= lim sup
ln ( , )
ln ( )
r
T r g
r↑
+
− −1
1
.
Dlq proyzvol\n¥x 0 1≤ ≤ ≤ +σ ρ σ opysan klass Aσ
ρ
analytyçeskyx v edynyçnom kruhe D =
= { }:z z < 1 funkcyj g z( ) takyx, çto g z( ) ≠ 0, ρ σT g[ ] = , ρ ρM g[ ] = , hde M r g( , ) =
= max ( ) :{ }g z z r≤ , T r u( , ) =
1
2
0
2
π
ϕ
π
ϕ∫ +ln ( )g re di , ρM g[ ] = lim sup
ln ln ( , )
ln ( )
r
M r g
r↑
+
− −1
1
,
ρT g[ ] = lim sup
ln ( , )
ln ( )
r
T r g
r↑
+
− −1
1
.
1. Vstup. Nexaj D = { z ∈ C : z < 1 } . Poznaçymo çerez A ( D ) ta H ( D )
klasy vidpovidno analityçnyx ta harmoniçnyx funkcij v D. Nexaj M ( r, f ) =
= max ( ) :{ }f z z r= , T ( r, f ) = 1
2 0
2
π
θθπ
log ( )+∫ f re di , de x+ = max ,{ }x 0 ,
0 < r < 1 i f ∈ A ( D ) — maksymum modulq ta xarakterystyka Nevanlinny vidpo-
vidno.
Porqdky zrostannq f ∈ D vyznaçagt\ tak:
ρM f[ ] = lim sup ln ln ( , )
ln( )r
M r f
r↑
+ +
− −1 1
, ρT f[ ] = lim sup ln ( , )
ln( )r
T r f
r↑
+
− −1 1
.
Vidomo, wo
ρT f[ ] ≤ ρM f[ ] ≤ ρT f[ ] + 1, (1)
i ci spivvidnoßennq utoçnyty ne moΩna.
Dlq zadanyx α > 1, ρ, ρ ≤ α ≤ ρ + 1, K. Linden [1] pobuduvav analityçnu v
D \ { }1 funkcig u vyhlqdi tak zvanoho dobutku Naftalevyça – Cudzi
g z( ) = E
a
a z
pn
nn
1
1
2
1
−
−
=
∞
∏ , , 1 1−( ) +∑ an
p
n
< ∞ ,
z vlastyvistg ρT g[ ] = ρ, ρM g[ ] = α. Tut
E ( w, p ) = ( ) exp{ / / }1 22− + + … +w w w w pp , p ∈ +Z ,
— pervynnyj mnoΩnyk Vej[rßtrassa, an — nuli g z( ).
Razom z tym problema povnoho opysu klasu Aσ
ρ
funkcij f ∈ A ( D ) takyx,
wo ρT g[ ] = σ, ρM g[ ] = ρ, dlq zadanyx ρ ≤ α ≤ ρ + 1 zalyßalas\ ne rozv’q-
zanog.
© I. E. ÇYÛYKOV, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7 979
980 I. E. ÇYÛYKOV
Osnovnym rezul\tatom statti [ povnyj opys pidklasu Aσ
ρ , qkyj sklada[t\sq
z analityçnyx funkcij bez nuliv u D. Cej rezul\tat otrymu[t\sq z opysu vidpo-
vidnoho klasu harmoniçnyx u D funkcij. Metod dovedennq spyra[t\sq na para-
metryçne zobraΩennq pidklasu H ( D ) funkcij skinçennoho porqdku ( ρM u[ ] <
< + ∞ ) , oderΩane M. M. DΩrbaßqnom [2], ta aparat drobovoho intehruvannq.
1.1. DopomiΩni vidomosti pro drobove intehruvannq. Dlq toho wob
sformulgvaty rezul\tat, nam potribni vidomosti z intehruvannq drobovoho
porqdku [2] (hl.CIX), [3] (hl.CXII.8). Dlq f ∈ L ( a, b ) ( intehrovno] za Lebehom na
( a, b ) ) drobovyj intehral Rimana – Liuvillq Fα porqdku α > 0 vyznaça[t\sq
formulog [2]
Fα
( r ) = D f r−α ( ) = 1
0
1
Γ( )
( ) ( )
α
α
r
r x h x dx∫ − − , r ∈ ( a, b ) ,
D
0
h
( r ) ≡ h
( r ) , D
α
h
( r ) = d
dr
D h r
p
p
p{ }( ) ( )− −α , α ∈ ( p – 1, p ] ,
de Γ( )α — hamma-funkciq. Fα neperervna pry α ≥ 1 i zbiha[t\sq z pervisnymy
vidpovidnoho porqdku pry α ∈ N . Zaznaçymo, wo pry α < 0 operator D
α
[
asociatyvnym ta komutatyvnym qk funkciq α.
Qkwo my ma[mo spravu z periodyçnymy funkciqmy, zokrema z tryhonomet-
ryçnymy rqdamy, to oznaçennq Rimana – Liuvillq [ nezruçnym. Tomu navedemo
we oznaçennq, wo naleΩyt\ H. Vejlg. Nexaj f ∈ L ( 0, 2π ) . Prypustymo, wo
0
2π
∫ f x dx( ) = 0. (2)
Qkwo c en
inx
n∈∑ Z
, c0 = 0, — rqd Fur’[ f, to drobovyj intehral (poxidna)
porqdku α vyznaça[t\sq rqdom Fur’[
c e
in
n
inx
n ( )\{ }
α
∈
∑
Z 0
α >
=
0
1
2
0
2
π
π
α∫ −f t x t dt( ) ( )Ψ , (3)
de Ψα( )t = e ininx
n
( )
\{ }
−
∈∑ α
Z 0
[ zbiΩnym majΩe skriz\ na [ 0, 2 π ] pry α > 0.
Tut i dali i
α = eiπα /2 . Intehral z (3) isnu[ majΩe skriz\, joho znaçennq
intehrovne, a rqd z (3) zbiΩnyj majΩe skriz\ i [ rqdom Fur’[ fα . Pry c\omu fα
zadovol\nq[ (2). Poznaçymo fα çerez I fα[ ]. Operator Iα ma[ ti sami vlas-
tyvosti, wo i D
α. Pry α ∈ ( 0, 1 ) poxidna f−α porqdku α vyznaça[t\sq for-
mulog f x−α( ) = d
dx
f x1−α( ). Qkwo Ω α > 0, n – 1 ≤ α < n, n ∈ N , to f x−α( ) =
= d
dx
f x
n
n n−α( ). MiΩ oznaçennqmy Rimana – Liuvillq ta H. Vejlq isnu[ takyj
zv’qzok:
f xα( ) = 1
2
0
2
π
π
α∫ −f x t t dt( ) ( )Ψ =
= 1 1
2
0
1
0
2
Γ( )
( )( ) ( ) ( )
α π
α
π
α
x
f t x t dt f t r x t dt∫ ∫− + −− ,
de r xα( ) — analityçna funkciq vid x, α > 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 981
Nexaj ψ : [ 0, 2 π ] → R . Çerez BV i AC poznaçymo klasy funkcij
obmeΩeno] zminy ta absolgtno neperervnyx na [ 0, 2 π ] vidpovidno. Qkwo
ψ β− ∈AC, α > 0, pysatymemo ψ β∈AC . Analohiçno ψ β∈BV , qkwo
ψ β− ∈BV. Vvedemo takoΩ poznaçennq f xα
∗( ) = i f xα
α( ) = c e nn
inx
n
−
≠∑ α
0
dlq f x( ) ~ c en
inx
n≠∑ 0
, α ∈ R . Nexaj
ω ( δ, ψ ) = sup ( ) ( ) : , [ , ],ψ ψ π δx y x y x y− ∈ − <{ }0 2
— modul\ neperervnosti funkci] ψ.
Qk i v [3], budemo hovoryty, wo ψ γ∈Λ , qkwo ω δ ψ( , ) = O( )δγ ( )δ↓0 .
Vplyv intehruvannq ta dyferencigvannq na modul\ neperervnosti opysu[ te-
oremaCCA.
Teorema))A. 1.CCNexaj 0 ≤ α < 1, β > 0, f ∈ Λ
α
. Todi:
a) fβ α β∈ +Λ pry α + β < 1;
b) fβ ∈Λ1 pry α + β > 1.
2.CCNexaj 0 < γ < α < 1. Todi:
a) f− −∈γ α γΛ pry f ∈Λα ;
b) f− −∈γ γΛ1 pry f ∈Λ1.
PunktyCC2,CC1a) vyplyvagt\ z teorem (8.13), (8.14) [3] (hl.CXII, teoremaC2),
punktCC1b) dovedemo v p.C3.
2. ZobraΩennq i zrostannq harmoniçnyx funkcij. Osnovni rezul\taty.
Dlq u ∈ H ( D ) oznaçymo maksymum modulq B ( r, u ) = max ( ) :{ }u z z r≤ ta xa-
rakterystyku Nevanlinny T ( r, u ) = 1
2 0
2
π
ϕϕπ
u re di+∫ ( ) . Vvedemo porqdky
ρB u[ ] = lim sup
ln ( , )
ln( )r
B r u
r↑
+
− −1 1
, ρT u[ ] = lim sup
ln ( , )
ln( )r
T r u
r↑
+
− −1 1
.
Podibno do (1) ma[mo spivvidnoßennq
ρT u[ ] ≤ ρM u[ ] ≤ ρT u[ ] + 1.
Nam znadoblqt\sq rozvynennq uzahal\nenyx qder Koßi, Ívarca ta Puassona:
C zα( ) =
Γ( )
( )
1
1 1
+
− +
α
αz
=
Γ
Γ
( )
( )
α + +
+=
+∞
∑ k
k
zk
k
1
10
,
(4)
S zα( ) = 2 0C z Cα α( ) ( )− , P rα ϕ( , ) = Re ( ) ( ){ }2 0C re Ci
α
ϕ
α− .
Zaznaçymo, wo P r0( , )ϕ = r D P r− −α α
α ϕ( ( , )) , P rα ϕ( , ) = D r P rα α ϕ( ( , ))0 , de
operator D di[ za zminnog r.
Nexaj Uα — pidklas H ( D ) takyx funkcij, wo
sup ( )
0 1 0
2
< <
∫
r
iu re d
π
α
ϕ ϕ = M
α < + ∞ ,
de u re i
α
ϕ( ) = r D u re i− −α α ϕ( ). Nexaj Uσ
ρ
— klas funkcij u ∈ H ( D ) takyx,
wo ρT u[ ] = σ, ρB u[ ] = ρ dlq zadanyx 0 ≤ ρ ≤ σ ≤ ρ + 1.
Teorema))B ([2], teoremaC9.10). Nexaj α > – 1. Todi u ∈ Uα todi i lyße
todi, koly
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
982 I. E. ÇYÛYKOV
u re i( )ϕ =
0
2π
α ϕ θ ψ θ∫ −P r d( , ) ( ), (5)
de ψ π∈BV[ , ]0 2 . Pry c\omu ψ θ( ) = lim ( )
n
n
iu r e d
→ +∞ ∫ α
θθ
θ
0
dlq deqko] posli-
dovnosti rn ↑1.
Z teori], pobudovano] M. M. DΩrbaßqnom [2] (hl. IX), moΩna otrymaty na-
stupne tverdΩennq, ne sformul\ovane nym u qvnomu vyhlqdi.
TverdΩennq. Nexaj u ∈ H ( D ) . Todi ρT u[ ] = inf :α α≥ ∈{ }0 u U .
Vyqvlq[t\sq, wo zrostannq B ( r, u ) , de u ma[ vyhlqd (5), opysu[t\sq v ter-
minax modulq neperervnosti ψ.
Teorema))C [4]. Nexaj α ≥ 0, 0 < γ < 1. Funkciq u ∈ H ( D ) zobraΩu-
[t\sq u vyhlqdi (5), de ψ γ∈BV ∩ Λ , todi i lyße todi, koly u U∈ α i
B ( r, u ) = O r(( ) )1 1− − −γ α , r↑1.
Slid zaznaçyty, wo dovedennq teoremyCCC podibne do dovedennq teoremyCCD,
vstanovleno] u vypadku, koly ψ ∈AC, u [5] (dyv. takoΩ [3], hl. VII, teore-
maC5.1).
Teorema))D. Nexaj u ∈ H ( D ) , 0 < γ ≤ 1. Todi
u re i( )ϕ =
0
2
0
π
ϕ∫ −P r t t dt( , ) ( )v
dlq deqko] v ∈ Λ
γ todi i lyße todi, koly
B r u, ∂
∂
ϕ
= O r(( ) )1 1− −γ , r↑1.
Analohy cytovanyx teorem [ pravyl\nymy i dlq analityçnyx funkcij. Slid
we zhadaty blyz\kyj do predmetu rozhlqdu dano] statti rezul\tat F. A. Íamo-
qna [6] (teoremaC3).
Teorema))E. Nexaj F ( z ) = exp ( ) ( )1
2 0
2
π
ψ θα
θπ
S ze di−∫{ }. Todi
sup ( ) ( , )
0 1 0
1
< <
−∫ −
r
r
r t T t F dtα < + ∞
todi i lyße todi, koly:
1) ψ ∈AC;
2)
0
2
0
2
2
2
π π ψ θ ψ θ ψ θ θ∫ ∫ + − + −( ) ( ) ( )t t
t
dt d < + ∞.
TeoremaCCE naßtovxu[ na dumku, wo za velyçynu porqdku ρT [ u ] , de u ma[
vyhlqd (5), vidpovidagt\ vlastyvosti ψ, pov’qzani z absolgtnog neperervnistg.
Ce, vlasne, vstanovlg[ teoremaC1, qka [ klgçovym rezul\tatom dano] statti.
Teorema))1. Nexaj u ma[ vyhlqd (5), α ≥ 0, ψ ∈BV. Todi:
1) ρT [ u ] = ( )α γ− +
1 , de γ1 = sup :{ }τ ψ τ≥ ∈0 AC ;
2) ρT [ u ] = ( )α γ− +
2 , de γ2 = sup :{ }τ ψ τ≥ ∈0 BV .
Zaznaçymo, wo γ1 = γ2 dlq dovil\no] funkci] ψ . Spravdi, nerivnist\ γ2 ≥
≥ γ1 [ oçevydnog, pozaqk ACτ ⊂ BVτ . Qkwo Ω ψ ∈BV, ψ ( x ) ∼ c en
inx
n≠∑ 0
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 983
to ψβ ( x ) = c n en
inx
n
−
≠∑ β
0
∈ AC dlq dovil\noho β > 0, tobto BVτ β+ ⊂ ACτ
dlq dovil\noho β > 0 [7, c. 31].
Nexaj τ [ ψ ] = sup :{ }γ ψ γ≥ ∈0 Λ , γ [ ψ ] = sup :{ }τ ψ τ≥ ∈0 AC .
Teorema))2. Nexaj 0 ≤ σ ≤ ρ ≤ σ + 1 < + ∞ , u ∈ H ( D ) . Funkciq u U∈ σ
ρ
todi i lyße todi, koly ( )∀ >α σ ( )∃ ∈ψ BV u ( z ) ma[ vyhlqd (5), σ =
= ( [ ])α γ ψ− +
i τ ψσ α[ ]− = 1 – ρ + σ.
Naslidok)1. Nexaj u ma[ vyhlqd (5), α ≥ 0, ψ ∈BV. Todi ρ T [ u ] =
= ( [ ])α γ ψ− + , ρB [ u ] = ρ τ ψρ αT uu
T
[ ] [ ][ ]–+ −1 .
ZauvaΩennq)1. Pry σ > 0 umova σ = ( [ ])α γ ψ− +
[ rivnosyl\nog umovi
γ [ ψ ] = α – σ.
ZauvaΩennq)2. Neobxidnist\ teoremyC2 ne ma[ miscq dlq α = σ. Rozhlqne-
mo funkcig u ( z ) = Re{ ( )}h z , h ( z ) = ( ) ln1 1
1
−
−
−z
z
ρ , de ρ ≥ 1, hilku mnoho-
znaçno] funkci] h ( z ) v D vybrano tak, wob h ( 1 / 2 ) = 2 2ρ ln . Oçevydno, wo
B ( r, u ) = ( ) ln1 1
1
−
−
−r
r
ρ , zokrema ρM [ g ] = ρ. Z ocinky dlq B ( r, u ) vyplyva[,
wo T ( r, u ) ≥ c r
r0
11 1
1
( ) ln−
−
− +ρ , r↑1, ale ρT [ u ] = ρ – 1. Z dovedennq lemyC3
vyplyva[, wo u ( z ) ne moΩna zobrazyty u vyhlqdi (5) z α = ρ – 1, bo todi b ma-
ly T ( r, u ) = O r(( ) )1 1− − +ρ , r↑1.
3. DopomiΩni tverdΩennq. Nastupna lema, qka ma[ i samostijnyj interes, [
osnovnog pry dovedenni teoremyC1.
Lema))1. Nexaj α > 0, β > – α – 1, ψ ∈AC ( pry β < 0 nexaj we
ψ β∈ −AC ) ,
g re i( )ϕ =
0
2π
α
ϕ ψ∫ −C re d ti t( ) ( )( ) . (6)
Todi
g re i( )ϕ =
0
2
1
π
α β
ϕ
β αβ
ϕψ∫ +
− ∗ +C re d t D Q rei t i( ) ( ) ( )( ) , (7)
de
Q zαβ( ) =
=
0
2
1 1
0
2
0 2
0
2
0
2
1
1
1 1
π
α β β
π
β
π
β αβ
π
α β
χ χ ψ χ χ α β
ψ ψ α β
∫
∫ ∫
∫
+ −
− ∗
− ∗ − ∗
+ −
= + ∈ + >
+ − + + =
C ze d t t t t
C ze d t D ze d t q z
C ze
it
it it
( ) ( ), ( ) ( ) ˜ ( ), ˜ , ,
( ) ( ) ln ( ) ( ) ( ), ,
(
Λ
−− ∗
− ∗
+ < + <
− + + =
− < + <
∫
it
it
d t q z
ze d t q z
q z
) ( ) ( ), ,
ln ( ) ( ) ( ), ,
( ), ,
ψ α β
ψ α β
α β
β αβ
π
β αβ
αβ
0 1
1 0
1 0
0
2
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
984 I. E. ÇYÛYKOV
q zαβ( ) — obmeΩena v D analityçna funkciq, D1
, D2 — stali, zaleΩni lyße
vid α i β.
Tut i dali hilku Ln w vybrano tak, wob ln 1 = 0.
MoΩna dovesty analohiçne tverdΩennq dlq harmoniçnyx funkcij, prote ob-
meΩymosq lyße okremym vypadkom, qkyj nam bude potriben dlq dovedennq teo-
rem.
Lema))2. Nexaj α > 0, β > – α , ψ ∈AC ( pry β < 0 nexaj we ψ β∈ −AC ) ,
u ( z ) ma[ vyhlqd (5). Todi
u re i( )ϕ =
0
2
1
π
α β β αβ
ϕϕ ψ∫ +
∗− + ′P r t d t D E re i( , ) ( ) ( ), (8)
de
E re i
αβ
ϕ( ) =
= 0
2
1 1
0
2
1
1
0 1
π
α β β
π
α β β αβ
ϕ χ χ ψ χ χ α β
ϕ ψ α β
∫
∫
+ −
∗
+ −
∗
− = + ∈ + >
− + < + <
P r t d t t t t
P r t d t e z
( , ) ( ), ( ) ( ) ˜ ( ), ˜ , ,
( , ) ( ) ( ), ,
Λ
e zαβ( ) — obmeΩena v D harmoniçna funkciq, ′D1 — stala, zaleΩna lyße vid
α i β.
Dovedennq lemy�1. Nexaj α + β > – 1. Oçevydno, wo
g ( 0 ) =
0
2
0
π
α ψ∫ C d t( ) ( ) = Γ( ) ( ) ( )( )α ψ π ψ+ −1 2 0 .
Poznaçymo ˜( ) ( ) ( )g z g z g= − 0 ,
˜ ( ) ( ) ( )C z C z Cα α α= − 0 . Todi ˜( )g re iϕ , ˜ ( )C rei
α
ϕ
zadovol\nqgt\ umovu (2) za zminnog ϕ.
ProdovΩymo ψ na R za formulog ψ π ψ( ) ( )t t+ −2 = ψ π ψ( ) ( )2 0− . Z (4) i
(6) oderΩu[mo
˜( )g re iϕ =
0
2π
α
ϕ ψ∫ −˜ ( ) ( )( )C re d ti t =
0
2π
α
θ ψ θ ϕ θ∫ − ′ +˜ ( ) ( )C re di . (9)
Za umovog lemy ψβ , a otΩe, i ψβ
∗
naleΩat\ do AC. Podi[mo operatorom Iβ na
(9). Za teoremog Fubini ma[mo
I g re i
β
ϕ[ ˜]( ) =
0
2π
α
θ
β ψ θ ϕ θ∫ − ′ +˜ ( ) [ ( )]C re I di =
= i C re di− − ∗∫ ′ +β
π
α
θ
βψ θ ϕ θ
0
2
˜ ( )( ) ( ) = i C re d ti t− − ∗∫β
π
α
ϕ
βψ
0
2
˜ ( ) ( )( ) =
= i
k
k
r e d t
k
k i t k−
=
+∞
− ∗∫ ∑ + +
+
β
π
ϕ
β
α ψ
0
2
1
1
1
Γ
Γ
( )
( )
( )( ) =
k
k
ikr e
=
∞
∑
1
µ ϕ( ) , (10)
de
µk r( ) = i
k
k
e d t ritk k− − ∗+ +
+ ∫β
π
β
α ψΓ
Γ
( )
( )
( )
1
1
0
2
.
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PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 985
Podi[mo teper na rivnist\ (10) operatorom I−β . OderΩu[mo
˜ ( )g re iϕ =
k
k
ikk r e
=
∞
∑
1
β ϕµ ( ) =
0
2
1
1
1
π
β ϕ
β
α ψ∫ ∑
=
+∞
− ∗+ +
+k
k i t kk
k
k
r e d t
Γ
Γ
( )
( )
( )( ) =
=
0
2
1
1
1
π
ϕ
β
α β ψ∫ ∑
=
+∞
− ∗+ + +
+k
k i t kk
k
r e d t
Γ
Γ
( )
( )
( )( ) +
+
0
2
1
1
1
1
1
π β
ϕ
β
α α β ψ∫ ∑
=
+∞
− ∗+ +
+
− + + +
+
k
k i t kk k
k
k
k
r e d t
Γ
Γ
Γ
Γ
( )
( )
( )
( )
( )( ) =
=
0
2
1
π
α β
ϕ
β
ϕψ∫ +
− ∗ +˜ ( ) ( ) ( )C re d t g rei t i , (11)
de
g re i
1( )ϕ =
0
2π
ϕ
βψ∫ − ∗G re d ti t( ) ( )( ) , G ( z ) =
k
k
kd z
=
+∞
∑
1
,
dk =
Γ Γ
Γ
( ) ( )
( )
k k k
k
+ + + − + +
+
α β αβ1 1
1
.
Z asymptotyky Γ ( z ) vyplyva[, wo dk = O k( )α β+ −1 , k → ∞ . Prote nam
potribna toçnißa ocinka. Dlq c\oho vykorysta[mo rqd Stirlinha [8, c. 40]
(hl.C12.33, teoremaC2)
Γ ( x ) = x e
x x
O
x
x x− − + + +
1 2
2 32 1 1
12
1
288
1/ π , x → + ∞ .
Nexaj y = O ( 1 ) , x → + ∞ , todi za dopomohog obçyslen\ (moΩna zastosuva-
ty paket Maple) oderΩymo
Γ
Γ
( )
( )
x y
x
+
= x
y y
x
y y y y
x
O
x
y 1
2
3 10 9 2
24
12 4 3 2
2 3+ − + − + − +
, x → + ∞ . (12)
Dlq znaxodΩennq asymptotyky dk zastosu[mo (12) z x = k + 1 , y ∈ +{ , }α α β . V
rezul\tati otryma[mo
dk = D k D k O k1
1
3
2 3α β α β α β+ − + − + −+ + ( ), k → + ∞ ,
de D1, D3 — stali, qki zaleΩat\ lyße vid α ta β.
Qkwo α + β < 0, to
k
k
kd r
=
+∞
∑
1
= O k
k=
+∞
+ −∑
1
1α β = O ( 1 ) , 0 ≤ r ≤ 1,
tobto G ( z ) , a otΩe i g1 ( z ) , — obmeΩena analityçna funkciq v D .
Qkwo α + β = 0, ma[mo dk = D k O k1
2/ ( )+ − , k → + ∞ . Zvidsy znaxodymo
G ( z ) = – D z g z1 31ln( ) ( )− + , de g3 — obmeΩena analityçna funkciq v D .
Pry α + β > 0 z ohlqdu na te, wo
Γ
Γ
( )
( )
k
k
+ +
+
α β
1
= k
K
k
K
k
O
k
α β+ − + + +
1 1 2
2 31 1 , k → + ∞ , (13)
oderΩu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
986 I. E. ÇYÛYKOV
dk = D
k
k
D
k
k
O k1 4
3
1
1
1
Γ
Γ
Γ
Γ
( )
( )
( )
( )
( )
+ +
+
+ + + −
+
+ + −α β α β α β , k → + ∞ , (14)
de K1, K2 , D4 — stali, qki zaleΩat\ lyße vid α ta β. Pry α + β < 1 ma[mo
dk = D
k
k
O k1
2
1
Γ
Γ
( )
( )
( )
+ +
+
+ + −α β α β , k → + ∞ .
Tomu G ( z ) = D C z1 1
˜ ( )α β+ − + g3 ( z ) , de g3 ( z ) — obmeΩena analityçna funkciq
vCC D .
Qkwo α + β = 1, to G ( z ) = D C z D z1 0 4 1˜ ( ) ln( )− − + g4 ( z ) , de g4 ( z ) — ob-
meΩena analityçna funkciq v D .
Pidsumovugçy dovedene vywe, pry – 1 < α + β ≤ 1 z (11) znaxodymo ˜( )g z =
= ˜ ( ) ( )C ze d tit
α β β
π
ψ+
− ∗∫0
2
+ g1 ( z ) , de g1 ( z ) ma[ vyhlqd, qkyj vymaha[t\sq v lemi
vid Q zαβ( ) . Mirkugçy, qk na poçatku dovedennq lemy, otrymu[mo
g ( z ) =
0
2π
α β βψ∫ +
− ∗C ze d tit( ) ( ) + g1 ( z ) + K3 ,
de K3 = – C gα β β βψ π ψ+
∗ ∗− +( )( ( ) ( )) ( )0 2 0 0 . OtΩe, u vypadku – 1 < α + β ≤ 1
lemu dovedeno.
Zalyßylos\ rozhlqnuty vypadok α + β > 1. Z (14) vyvodymo
G ( z ) =
k
k
kd z
=
+∞
∑
1
= D C z D C z z
k
k
k
1 1 4 2
1
˜ ( ) ˜ ( )α β α β+ − + −
=
+∞
+ + ∑ ∆ ,
∆k = O k( )α β+ −3 , k → + ∞ .
OtΩe,
g1 ( z ) = D C ze d t D C ze d tit it
1
0
2
1 4
0
2
2
π
α β β
π
α β βψ ψ∫ ∫+ −
− ∗
+ −
− ∗+˜ ( ) ( ) ˜ ( ) ( ) +
+
0
2
1
π
βψ∫ ∑
=
+∞
− ∗
k
k
k itkz e d t∆ ( ) ≡
≡ D C ze d t h z h zit
1
0
2
1 1 2
π
α β βψ∫ + −
− ∗ + +˜ ( ) ( ) ( ) ( ).
Po-perße, z vyhlqdu h 1 ( z ) vyplyva[ M ( r, h1 ) = O r(( ) )1 1− − −β α
pry r↑1.
Po-druhe, z rezul\tativ M. M. DΩrbaßqna [2] (hl. 9, § 4) vyplyva[, wo
h R1 2∈ + −α β ⊂ Rα β+ −1, de Rα — klas analityçnyx u D funkcij takyx, wo
Re f U∈ α . Tomu [2] (hl.CIX)
h1 ( z ) =
0
2
1 1 0
π
α β χ∫ + −
− +S ze d t i hit( ) ( ) Im ( ),
de χ1 ∈BV .
Rozhlqnemo h2 ( z ) . Pry r↑1
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PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 987
M ( r, h2 ) = O k r
k
k
=
+∞
+ −∑
1
3α β =
O
r
O
r
1
1
2
1
1
2
2( )
, ,
ln , .
( )−
+ ≠
−
+ =
+ − +α β
α β
α β
Zvidsy lehko otrymaty, wo D h re i− − +α β ϕ1
2Re ( ) = O ( 1 ) , r↑1. Spravdi, ce oçe-
vydno pry α + β < 2. A pry α + β ≥ 2, 1 / 2 > ε > 0
D h re i− − +α β ϕ1
2Re ( ) = r r t h te dt
r
i
− − +
+ −
+ −
−∫
α β
α β ϕ
α β
1
0
2
21Γ ( )
( ) Re ( ) =
= O
r t dt
t
r
0
2
21∫ −
−
+ −
+ − +
( )
( )
α β
α β ε = O dt
r t
r
0
∫ −
( ) ε = O ( 1 ) , r↑1.
OtΩe, h R2 1∈ + −α β i
h2 ( z ) =
0
2
1 2 2 0
π
α β χ∫ + −
− +S ze d t i hit( ) ( ) Im ( ) ,
de χ2 ∈BV. Ale oskil\ky M r h h( , )1 2+ = O r(( ) )1 1− − −α β , r↑1, to
( )( )h h z1 2+ =
0
2
1 1 2 1 20 0
π
α β χ χ∫ + −
− + + +S ze d t t i h hit( ) ( ( ) ( )) Im( ( ) ( )) =
=
0
2
1
π
α β χ∫ + −
−C ze d tit( ) ˜ ( ) ,
de ˜ ( )χ t = 2 1 2 4( ( ) ( ))χ χt t K t+ + , K4 — stala. Zastosuvavßy teoremuCCC do
Re( )h h1 2+ , otryma[mo, wo χ χ1 2 1+ ∈Λ i, qk naslidok, χ̃ ∈Λ1. Zvidsy i z (11)
vyplyva[ tverdΩennq lemy u vypadku α + β > 1.
Lemu dovedeno.
Dovedennq lemy�2. Nexaj u ( z ) ma[ vyhlqd (5),
F ( z ) =
0
2π
α ψ∫ −S ze d tit( ) ( ) =
0
2
52
π
α ψ α∫ − −C ze d t Kit( ) ( ) ( ).
Todi Re F = u. Zastosovugçy do g ( z ) = ( ( ) ) /F z K+ 5 2 lemuC1, oderΩu[mo
1
2 5( ( ) )F z K+ =
0
2
1
π
α β β α βψ∫ −
−
−
∗
−+C ze d t D Q zit( ) ( ) ( ), ,
de R zα β, ( )− magt\ vyhlqd, opysanyj u lemiC1. Zvidsy
F ( z ) =
0
2
1 62
π
α β β α βψ∫ −
−
−
∗
−+ −S ze d t D Q z Kit( ) ( ) ( ), .
OtΩe,
u re i( )ϕ =
0
2
1 62
π
α β β α β
ϕϕ ψ∫ − −
∗
−− + −P r t d t D Q re Ki( , ) ( ) Re( ( ) ), .
Vraxovugçy, wo P r tα ( , ) = Re( ( ) ( ))2 0C re Cit
α α− , pry 0 < α + β < 1 otry-
mu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
988 I. E. ÇYÛYKOV
E re i
αβ
ϕ( ) = 2 1 6D Q re KiRe( ( ) ),α β
ϕ
− − = D P r t d t q z1
0
2
1
π
α β βϕ ψ∫ − − −
∗− +( , ) ( ) Re ( ),
de q ( z ) — obmeΩena v D analityçna funkciq.
OtΩe, u vypadku 0 < α + β < 1 lemu dovedeno. Pry α + β > 1 ma[mo
( ˜ , )χ ∈ ∈Λ1 7K R
E re i
αβ
ϕ( ) = D P r t d t K1
0
2
1 7
π
α β ϕ χ∫ − − − +( , ) ( ) = D P r t d t1
0
2
1
π
α β ϕ χ∫ − −
∗−( , ) ( ),
de χ∗( )t = χ π α β( ) ( ( ))/t K D P t+ − −7 1 12 0 .
Lemu dovedeno.
Lema))3. Qkwo u ( z ) ma[ vyhlqd (5), ψ β∈AC , to
T ( r, u ) = O
r
1
1( )( )−
− +α β
, r↑1.
Dovedennq. MoΩemo vvaΩaty, wo α ≥ β. Nexaj spoçatku α > β. Za le-
mogC2 ma[mo
u re i( )ϕ =
0
2
1
0
2
1
π
α β β
π
α βϕ ψ ϕ χ∫ ∫− −
∗
− −− + ′ −P r t d t D P r t d t( , ) ( ) ( , ) ( ) ≡ I I1 2+ , (15)
de χ ∈BV . Dali zi standartnyx ocinok znaxodymo
T ( r, I1 ) = 1
2
0
2
0
2
π
ϕ ψ ϕ
π π
α β β∫ ∫ − −
∗
+
−
P r t d t d( , ) ( ) ≤
≤ 1
2
0
2
0
2
π
ϕ ϕ ψ
π π
α β β∫ ∫ − −
∗−
P r t d d t( , ) ( ) ≤
≤ 1
2
2 1
10
2
0
2
1π
α β ϕ ψ
π π
ϕ α β β∫ ∫ +
−
− − + −
∗Γ ( – )
( )
( )re
d d t
i t
≤
≤
Γ ( – )
( )
1
10
2
1
1
+
−
∫ ∫ − +
− ≤ −
α β
π
ϕπ
α β
ϕ
d
rt r
+
+
d
r t
d d t
r t t
ϕ
ϕ
ϕ ψα β
ϕ π π ϕ π
β( sin( ))
( )
/ /−
+
− +
− < − ≤ < − ≤
−
∗∫ ∫1
1 2 2
≤
≤
Γ ( – )
( ) ( / )
/
1 2
1
2
21
1
2
1
+
−
+ +
− − +
−
− +∫α β
π
τ
τ π
πα β α β
π
α βr r
d
r
=
= O r( )1 −( )− +β α , r↑1.
Pry α = β, vykorystovugçy nevid’[mnist\ qdra Puassona, ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 989
T ( r, I1 ) = 1
2
0
2
0
2
0π
ϕ ψ ϕ
π π
β∫ ∫ −
−
∗
+
P r t d t d( , ) ( ) ≤
≤ 1
2
0
2
0
2
0π
ϕ ϕ ψ
π π
β∫ ∫ −
−
∗P r t d d t( , ) ( ) =
0
2π
βψ∫ −
∗d t( ) ≤ K,
de K — dodatna stala. OtΩe,
T ( r, I1 ) = O
r
1
1( )( )−
− +α β
, r↑1.
Analohiçno
T ( r, I2 ) = O
r
1
1 1( )( )−
− − +α β
, r↑1.
Ostatoçno
T ( r, u ) = O
r
1
1( )( )−
− +α β
, r↑1.
Lemu dovedeno.
Lema))4. Nexaj χ µ∈BV ∩ Λ , µ ∈ ( 0, 1 ] , p > 0. Todi
0
2
1 1 1
0
π
ϕ χ∫ − − − − −r D r P r t d tp p( )( , ) ( ) = O ( 1 ) , r↑1.
Dovedennq. Ma[mo
I
df=
0
2
1 1 1
0
π
ϕ χ∫ − − − − −r D r P r t d tp p( )( , ) ( ) =
= r s P s t ds d tp
r
p− − −∫ ∫ −1
0
2
0
1
0
π
ϕ χ( , ) ( )) =
= r s P s t d t dsp
r
p− − −∫ ∫ −1
0
1
0
2
0
π
ϕ χ( , ) ( )) .
Oskil\ky χ µ∈Λ , za teoremogCCC otrymu[mo
0
2
0
π
ϕ χ∫ −P s t d t( , ) ( )) = O s( )1 1−( )− +µ , s↑1.
Tomu dlq deqkoho r0 ∈ ( 0, 1 )
B ( r, I ) = O O s
s
ds
r
r p
( )
( )
1
1
0
1
1+
−
∫
−
−µ = O ( 1 ) , r↑1.
Lemu dovedeno.
Dovedennq tverdΩennq. Nexaj β = inf :{ }α α≥ ∈0 u U , α 1 > β. Todi, os-
kil\ky U Uα α1
⊃ , α1 > α, ma[mo u U∈ α1
. Za teoremog DΩrbaßqna [2] (teore-
maC9.10)
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990 I. E. ÇYÛYKOV
u re i( )ϕ =
0
2
1
π
α ϕ θ ψ θ∫ −P s d( , ) ( )) , (16)
de ψ ∈BV. Dali zi standartnyx ocinok oderΩu[mo T ( r, u ) = O r( )( )1 1− −α ,
r↑1. OtΩe, ρ α[ ]u ≤ 1, vidtak ρ β[ ]u ≤ .
Navpaky, prypustymo, wo T ( r, u ) = O r( )( )1 − −γ , r↑1, dlq deqkoho γ ∈ ( 0,
β ) . Nahada[mo deqki xarakterystyky M. M. DΩrbaßqna [2] ( )( )u H∈ D :
Tα ( r, u ) = r D u re di
−
− +
∫ ( )
α π
α ϕ
π
ϕ
2
0
2
( ) = 1
2
0
2
π
ϕ
π
α
ϕ∫ ( )+
u re di( ) .
Pry α > 0 vykonu[t\sq nerivnist\ Tα ( r, u ) ≤ r D T r u− −α α ( , ).
Z naßoho prypuwennq dlq α ∈ ( γ, β ) ma[mo D T r u−α ( , ) = O ( 1 ) , tomu
Tα ( r, u ) = O ( 1 ) . Oskil\ky uα [ harmoniçnog, ce rivnosyl\no tomu, wo
sup ( )
0 1
2
0
< <
∫
r r
iu re d
π
α
ϕ ϕ < + ∞ ,
tobto u U∈ α , wo supereçyt\ nerivnosti α < β. OtΩe, naße prypuwennq [ xyb-
nym, tobto ρ [ u ] ≥ β.
Ostatoçno ρ [ u ] = β.
TverdΩennq dovedeno.
Dovedennq punktu 1b) teoremy A. Budemo mirkuvaty, qk i pry dovedenni
(8.13) [3, c. 204] (teoremaC2). Vidomo [3, c. 204] (8.15), wo Ψα zadovol\nq[ neriv-
nosti
Ψα( )t ≤ c tα
α−1, Ψα( )t ≤ c tα
α−2 , 0 < t ≤ π,
de stala cα zaleΩyt\ lyße vid α .
Nexaj f ∈Λα , 0 < α < 1, 0 < β ≤ 2, 0 < h ≤ π / 2 . Todi
2π β β( )( ) ( )f x h f x+ − =
=
t h h t
f x t f x t h t dt
≤ ≤ ≤
∫ ∫+
− − + −
2 2 π
β β( ) ( )( ) ( ) ( ) ( )Ψ Ψ ≡ A + B .
Qk i v [3, c. 204], ma[mo
A = O t t h t dt
h
h
−
∫ + +( )
2
2
α
β βΨ Ψ( ) ( ) ≤
≤ O h t dt
h
h
( ) ( )α
β
−
∫
3
3
2 Ψ = O h t dt
h
( )α β
0
3
1∫ − = O h( )α β+ , h↓0 .
Vykorystovugçy teoremu pro seredn[ ta ocinku dlq ′Ψβ , otrymu[mo ( 0 < θ < 1 )
B = O t t h hdt
h t2 ≤ ≤
∫ ′ +
π
α
β θΨ ( ) ≤
≤ O h t t h dt
h t
( )
2
2
≤ ≤
−∫ −( )
π
α β = O h t dt
h
( )
π
α β∫ + −2 = O h( ) , h↓0 .
OtΩe, f x h f xβ β( ) ( )+ − = O h( ) pry h↓0 , tobto fβ ∈Λ1.
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PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 991
4. Dovedennq teorem. Dovedennq teoremy�1. Z lemyC3 vyplyva[, wo
ρT [ u ] ≤ ( )α γ− + . OtΩe, dosyt\ dovesty, wo ρT [ u ] ≥ ( )α γ− + . Oskil\ky pry
α ≤ γ nerivnist\ tryvial\na, moΩemo vvaΩaty, wo α > γ . Prypustymo, wo
ρT [ u ] = σ < α – γ . Nexaj η > 0 take, wo ρ = σ + η < α – γ . Za tverdΩen-
nqmC1
u re i( )ϕ =
0
2π
ρ ϕ λ∫ −P r t d t( , ) ( ), λ ∈ BV . (17)
Bil\ß toho, λ ∈ AC , oskil\ky v protyleΩnomu vypadku, qkwo rozhlqnuty ana-
lityçnu v D funkcig f taku, wo Re f = u, za teoremogCCE matymemo ρT [ u ] ≥
≥ ρ > σ, wo nemoΩlyvo.
Nexaj spoçatku α – γ ≥ 1, ε > 0. Za lemogC2 z (5) otrymu[mo ( β = – γ + ε )
u re i( )ϕ =
0
2
0
2
1
π
α γ ε ε γ
π
α γ εϕ ψ ϕ ψ∫ ∫− + −
∗
− − +− + −P r t d t P r t d t( , ) ( ) ( , ) ˜ ( ) ,
de ψ̃ ψ χε γ= +−
∗c1 1, χ1 1∈Λ , c1 — deqka dijsna stala. Znovu za lemogC2 z (17)
ma[mo
u re i( )ϕ =
0
2
0
2
1
π
α γ ε α ε γ ρ
π
α γ εϕ λ ϕ λ∫ ∫− + + − −
∗
− − +− + −P r t d t P r t d t( , ) ( ) ( , ) ˜ ( ),
de λ̃ λ χα ε γ ρ= ++ − +
∗c2 2, χ2 1∈Λ , c2 — deqka dijsna stala.
Poznaçymo τ = α – γ + ε – ρ > ε. VvaΩa[mo, wo τ < 1. C\oho moΩna dosqh-
nuty vyborom η. Dovedemo, wo ψ γ τ ε∈ + −AC 1 , ε1 < τ. Ce supereçytyme prypu-
wenng teoremy.
Z ostannix dvox zobraΩen\ u vyvodymo
0
2π
α γ ε ε γ τϕ ψ λ∫ − + −
∗ ∗− −P r t d t t( , ) ( ( ) ( )) =
0
2
1
π
α γ ε ϕ λ ψ∫ − + − − −P r t d t t( , ) ( ˜ ( ) ˜ ( )),
de
˜ ˜λ ψ− = c c2 1 2 1λ ψ χ χτ ε γ
∗
−
∗− + − . Zastosu[mo operator r D− −β β
do obydvox
çastyn ostann\o] rivnosti z β = α – γ + ε. Pry c\omu vraxu[mo, wo r D P− −β β
β =
= P0 ta rivnosti D−β = D D− − +1 1( )β , D P− +
−
β
β
1
1 = r Pβ−1
0 . OderΩymo
v( )re iϕ
df=
0
2
0
π
ε γ τϕ ψ λ∫ − −−
∗ ∗P r y d t t( , ) ( ( ) ( )) =
=
0
2
1 1
0
π
α γ ε α γ ε ϕ λ ψ∫ − + − − − + − − −r D r P r t d t t( )( , ) ( ˜ ( ) ˜ ( )).
Funkciq v harmoniçna v D i
′vr
ire( )ϕ = ( ) ( , ) ( ˜ ( ) ˜ ( ))( )− + − − −∫ − + − − − − + −α γ ε ϕ λ ψ
π
α γ ε α γ ε
0
2
1 1 1
0r D r P r t d t t +
+
0
2
1
0
π
ϕ λ ψ∫ − − −r P r t d t t( , ) ( ˜ ( ) ˜ ( ))) ≡ I I3 4+ . (18)
Oskil\ky λ ∈ ⊂AC Λ0 i 0 < ε < τ, to za teoremogCCA (1 a)) λτ τ ε
∗ ∈ ⊂Λ Λ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
992 I. E. ÇYÛYKOV
Za prypuwennqm teoremy ψ γ ε∈ −AC 0
dlq dovil\noho ε0 > 0, tobto
ψε γ
ε ε
−
∗ −∈AC 0 , 0 < ε0 < ε. OtΩe, dlq dovil\noho ε0 ∈ ( 0, ε ) ma[mo ψε γ−
∗
∈
∈ Λε ε− 0
. Z oznaçen\ λ̃ ta ψ̃ vyplyva[, wo
λ̃ – ψ̃ ∈ Λε ε− 0
. (19)
Z (19) za teoremogCCC vyplyva[, wo B ( r, I4 ) = O r(( ) )1 0 1− − −ε ε , r↑1.
Z inßoho boku, za lemogCC4 B ( r, I3 ) = O ( 1 ) , r↑1. Z ocinok I3 ta I4 vyvo-
dymo B r r( , )′v = O r(( ) )1 1 0− − + −ε ε
, r↑1. Za teoremogC2.35 [3] (hl.CVII.2] taka Ω
ocinka ma[ misce i dlq ′vϕ , a same B r( , )′vϕ = O r(( ) )1 1 0− − + −ε ε
, r↑1. Z ostan-
n\oho spivvidnoßennq i teoremyCCD vyplyva[, wo v zobraΩu[t\sq intehralom
Puassona vid funkci] z klasu Λε ε− 0
. Z inßoho boku, v [ intehralom Puassona
vid ( )χ∗ ′ , qka vyznaçena majΩe skriz\, de χ∗ = ψ λε γ τ−
∗ ∗− . Tobto ( )χ∗ ′ majΩe
skriz\ dorivng[ neperervnij funkci] z Λε ε− 0
. OtΩe, moΩemo vvaΩaty, wo
( )χ∗ ′ ∈ Λε ε− 0
. Zvidsy za teoremogCCA (1 b)) χ∗ ∈Λ1.
Ale pryhada[mo, wo λτ
τ
τ
∗ ∈ ⊂AC Λ . OtΩe, ψ χ λε γ τ τ−
∗ ∗ ∗= + ∈Λ . Zvidsy
˜ ˜λ ψ− = c c2 1 2 1λ ψ χ χτ ε γ τ
∗
−
∗− + − ∈Λ .
Tak samo, qk i vywe, ocinymo I4 , vykorystovugçy spivvidnoßennq
˜ ˜λ ψ− ∈
∈ Λτ zamist\
˜ ˜λ ψ− ∈ Λε ε− 0
.
Za teoremogCCC oderΩymo I4 = O r(( ) )1 1− −τ , a za lemogCC4 I3 = O ( 1 ) , i, qk
naslidok, B r( , )′vϕ = O r(( ) )1 1− −τ
pry r↑1.
Teper zhidno z teoremog Xardi – Littlvuda (teoremogCD) ma[mo ( )χ∗ ′ ∈ Λτ
abo ψ λε γ τ− −
∗
−
∗−1 1 ∈ Λτ . Rozhlqnemo intehral Vejlq porqdku 1 – τ + ε2
, 0 <
< ε2 < τ – ε , vid ostann\o] funkci]. Za teoremogCCA (1 b)) ψ λε γ τ ε ε− − +
∗ ∗−
2 2
∈
∈ Λ1 ⊂ AC . Ale λ ∈AC, tym bil\ße λε2
∗ ∈AC, otΩe, takog Ω [ funkciq
ψε γ τ ε− − +
∗
2
. Tobto ψ γ τ ε ε∈ + − −AC 2 . Ce supereçyt\ prypuwenng teoremy,
oskil\ky τ – ε – ε2 > 0.
Nexaj teper 0 < α – γ < 1. Vybyra[mo ε > 0 tak, wob 0 < α – γ + ε < 1.
Zastosovugçy, qk i vywe, lemuCC2 do zobraΩen\ (5) i (17), oderΩu[mo
u re i( )ϕ =
0
2π
α γ ε ε γϕ ψ∫ − + −
∗−P r t d t( , ) ( ) +
+ c P r t d t u re i
1
0
2
1 1
π
α γ ε ε γ
ϕϕ ψ∫ − − + −
∗− +( , ) ( ) ( ),
u re i( )ϕ =
0
2π
α γ ε α ε γ ρϕ λ∫ − + + − −
∗−P r t d t( , ) ( ) +
+ c P r t d t u re i
2
0
2
1 2
π
α γ ε α ε γ ρ
ϕϕ λ∫ − − + + − −
∗− +( , ) ( ) ( ) ,
de cj — stali, uj — obmeΩeni v D harmoniçni funkci], j ∈ { 1, 2 } . Zvidsy ( τ =
= α – γ + ε – ρ )
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 993
0
2π
α γ ε ε γ τϕ ψ λ∫ − + −
∗ ∗− −P r t d t t( , ) ( ( ) ( )) =
=
0
2
1 2 1 1 2
π
α γ ε τ ε γ
ϕϕ λ ψ∫ − + −
∗
−
∗− − + −P r t d c t c t u u re i( , ) ( ( ) ( )) ( )( ). (20)
Oskil\ky u2 – u1 — obmeΩena harmoniçna funkciq v D, ]] moΩna zobrazyty
(teoremaCCC) u vyhlqdi
( )( )u u re i
1 2− ϕ =
0
2
0
π
ϕ χ∫ −P r t d t( , ) ( ),
de χ ∈Λ1.
Zastosu[mo operator r D− −β β
do obox çastyn rivnosti (20) z β = α – γ + ε.
Qk i u vypadku α – γ ≥ 1, oderΩymo
v( )re iϕ
df=
0
2
0
π
ε γ τϕ ψ λ∫ − −−
∗ ∗P r t d t( , ) ( )( ) =
=
0
2
1 1
0 2 1
π
α γ ε α γ ε
τ ε γϕ λ ψ∫ − + − − − + − ∗
−
∗− −r D r P r t d c t c t( )( , ) ( ( ) ( )) +
+
0
2π
α γ ε ϕ χ∫ − + − −P r t d t( , ) ( ).
Dyferenciggçy ostanng rivnist\, otrymu[mo
′vr
ire( )ϕ =
0
2
1
0 2 1
π
τ ε γϕ λ ψ∫ − ∗
−
∗− −r P r t d c t c t( , ) ( ( ) ( )) +
+ ( ) ( , ) ( ( ) ( ))( )− + − − −∫ − + − − − − + − ∗
−
∗α γ ε ϕ λ ψ
π
α γ ε α γ ε
τ ε γ
0
2
1 1 1
0 2 1r D r P r t d c t c t +
+
0
2π
α γ ε ϕ χ∫ ∂
∂
−− + −r
P r t d t( , ) ( ) ≡ I I I5 6 7+ + . (21)
I5 ta I6 ocinggt\sq tak samo, qk i u vypadku α – γ > 1. Ma[mo λτ τ
∗ ∈Λ ,
ψε γ ε ε−
∗
−∈Λ
0
, 0 < ε0 < ε . Za teoremogCCC ta lemogCC4 vidpovidno
B ( r, I5 ) = O r(( ) )1 1 0− − + −ε ε , B ( r, I6 ) = O ( 1 ) , r↑1. (22)
I7 ocing[mo standartno [3] (hl.C8.2). Spoçatku zaznaçymo, wo
∂
∂ ∂ − + −
2
t r
P r tα γ ε ( , ) ≤
K
reit
8
3
1 −
− + − +α γ ε . (23)
Zvidsy, zokrema, ma[mo
∂
∂ ∂ − + −
2
t r
P r tα γ ε ( , ) ≤
K
t
9
3− + − +α γ ε , t ≤ π. (24)
ProdovΩymo χ na R za formulog χ π χ( ) ( )t t+ −2 = χ π χ( ) ( )2 0− . Os-
kil\ky
∂
∂ − + −r
P r tα γ ε ( , ) ta χ ′ — periodyçni funkci] vid t, oderΩu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
994 I. E. ÇYÛYKOV
I7 =
− +
+
− + −∫ ∂
∂
− −
π ϕ
π ϕ
α γ ε θ ϕ χ θ χ ϕ
r
P r d( , ) ( ( ) ( )) =
= ( ( ) ( )) ( , )χ θ χ ϕ θ ϕα γ ε
π ϕ
π ϕ
− ∂
∂
−− + −
− +
+
r
P r –
–
− +
+
− + −∫ ∂
∂
∂
∂
−
−
π ϕ
π ϕ
α γ εθ
θ ϕ χ θ χ ϕ θ
r
P r d( , ) ( ( ) ( )) =
= ( ( ) ( )) ( , ) ( , ) ( ( ) ( ))χ π χ π
τ
χ ϕ χ ϕα γ ε
π
π
α γ ε2 0− ∂
∂
− ∂
∂
∂
∂
+ −− + −
−
− + −∫r
P r
r
P r s s ds .
Zvidsy, vykorystovugçy ocinky qdra (23), (24), vyvodymo
I7 ≤ K10( )χ +
τ τ π
α γ ετ
τ ω τ χ τ
≤ − − ≤ ≤
− + −∫ ∫+
∂
∂
∂
∂
1 1r r
r
P r d( , ) ,( ) ≤
≤ K10 + K
r
d
r
11
1
31τ
α γ ε
ω τ χ τ
≤ −
− + − +∫ −
( ),
( )
+
1
12
3
− ≤ ≤
− + − +∫
r
K
d
τ π
α γ ετ
ω τ χ τ( ), ≤
≤ K10 +
2 1
1
11
2
K r
r
ω χ
α γ ε
( ),
( )
−
− − + − + + K d
r
13
1
2
− ≤ ≤
− + − +∫
τ π
α γ ε
τ
τ
=
= O
r
1
1 1( )−
− + −α γ ε = O
r
1
1 1( )−
− −τ ρ , r↑1.
Razom z (22) ce da[ B r r( , )′v = O r(( ) )1 1 0− − + −ε ε
, r↑1, ε0 ∈ ( 0, ε ) . Dali mirku-
vannq taki, qk u vypadku α – γ > 1 vid momentu zastosuvannq teoremyCCD.
TeoremuCC1 dovedeno.
Dovedennq teoremy�2. Dostatnist\. Nexaj α > σ. Za teoremogCC1
ρ [ u ] = ( [ ])α γ ψ− + = σ.
Prypustymo spoçatku, wo σ < ρ. Vyberemo α ∈ ( σ, ρ ) . Oskil\ky τ ψσ α[ ]− =
= 1 – ρ – σ, ψσ α δ− ∈Λ pry δ < 1 – ρ + σ. Za teoremog A (1 a)) ψ γ∈Λ pry
γ = δ + α – σ < 1 – ρ + α . Vodnoças ψ γ∉Λ pry γ > 1 – ρ + α , bo v proty-
leΩnomu vypadku za teoremogCCA maly b τ ψσ α[ ]− > 1 – ρ + σ. OtΩe, τ [ ψ ] =
= 1 – ρ + α. Za teoremogCCC ρB [ u ] = ρ.
Nexaj teper σ = ρ. Dosyt\ dovesty, wo ρB [ u ] ≤ σ. Nexaj α [ dovil\nym
bil\ßym za σ. Oskil\ky τ ψσ α[ ]− = 1, za teoremogCCA (1 b)) ma[mo ψ ∈Λ1.
Zvidsy za teoremogCCC B ( r, u ) = O r(( ) )1 − −α , tobto ρB [ u ] ≤ α . Dostatnist\
dovedeno.
Neobxidnist\. Nexaj ρB [ u ] = ρ, ρT [ u ] = σ, σ ≤ ρ ≤ ρ + 1. Za tverdΩen-
nqm dlq dovil\noho α > σ funkcig u moΩna zobrazyty u vyhlqdi (5), pry c\o-
mu za teoremogCC1 ma[mo ( [ ])α γ ψ− + = σ.
Prypustymo spoçatku, wo σ < ρ. Nexaj α ∈ ( σ, ρ ] . Z oznaçennq ρB vyply-
va[, wo B ( r, u ) = O r(( ) )1 − − −ρ ε , r↑1, dlq dovil\noho ε > 0. Nexaj 0 < ε ≤
≤ 1 – ρ + α . Za teoremogCCC ma[mo ψ α ρ ε∈ + − −Λ 1 . Todi ψσ α σ ρ ε− + − −∈Λ 1 .
Krim toho, ψσ α σ ρ ε− + − +∉Λ 1 dlq dodatnoho ε, bo v protyleΩnomu vypadku ψC∈
∈ Λα ρ ε+ − +1 i za teoremogCCC maly b B ( r, u ) = O r(( ) )1 − − +ρ ε , r↑1, ale
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PRO POVNYJ OPYS KLASU ANALITYÇNYX U KRUZI FUNKCIJ BEZ NULIV … 995
B ( rn ,Cu ) ≠ O rn(( ) )1 − − +ρ ε
na deqkij poslidovnosti rn ↑1. OtΩe, τ ψσ α[ ]− = 1 –
– ρ + σ.
Nexaj teper α > ρ > σ, u zobraΩu[t\sq u vyhlqdi (5). Za lemogC2 dlq βC∈
∈ ( σ, ρ ) , χ ∈ BV ma[mo
u re i( )θ =
0
2
0
2
1
π
β β α
π
βϕ ψ ϕ χ∫ ∫− + −−
∗
−P r t d t P r t d t( , ) ( ) ( , ) ( ) = I I8 9+ . (25)
Oskil\ky B ( r,CI9 ) = O r(( ) )1 − −β , r↑1, a ρB [ u ] = ρ, to ρB [ I8 ] = ρ.
Za teoremogCCC τ ψβ α[ ]−
∗ = τ ψβ α[ ]− = β + 1 – ρ . Zvidsy za teoremogCCA
(2 a)) τ ψσ α[ ]− = σ + 1 – ρ .
Nareßti, nexaj ρ = σ < α , ε > 0, β = σ + ε < α . Za lemogCC2 znovu ma[mo
(25). Zi spivvidnoßen\ ρB [ u ] = ρ , ρ B [ I9 ] ≤ β otrymu[mo ρB [ I8 ] ≤ ρ + ε .
OtΩe, B ( r, I8 ) = O r(( ) )1 2− − −ρ ε , r↑1. Za teoremogCCC τ ψβ α[ ]− ≥ β – 1 – ρ –
– 2ε = 1 – ε. Zvidsy za teoremogCCA (2 a)) τ ψσ α[ ]− ≥ 1 – 2ε. Z dovil\nosti ε > 0
vyplyva[ τ ψσ α[ ]− = σ + 1 – ρ .
TeoremuCC2 dovedeno.
5. Naslidky dlq analityçnyx funkcij. Z teoremC1, 2 ta toho faktu, wo
dlq analityçno] funkci] g bez nuliv ln ( )g z — harmoniçna funkciq, vyplyva-
gt\ taki teoremy.
Teorema))3. Nexaj g z( ) = exp{ ( )}h z , de
h z( ) =
0
2
0
π
α ψ∫ − +S ze d t i hit( ) ( ) Im ( ), (26)
α ≥ 0, ψ ∈ BV . Todi ρT g[ ] = ( [ ])α γ ψ− + .
Teorema))4. Nexaj 0 ≤ σ ≤ ρ ≤ σ + 1 < + ∞ , g — analityçna funkciq i
g z( ) ≠ 0 v D . Funkciq g A∈ σ
ρ
todi i lyße todi, koly ( )∀ >α σ ( )∃ ∈ψ BV
g z( ) = exp{ ( )}h z , de h z( ) ma[ vyhlqd (26), σ = ( [ ])α γ ψ− +
i τ ψσ α[ ]− = 1 –
– ρ + σ.
Naslidok)2. Nexaj g z( ) = exp{ ( )}h z , de h ( z ) ma[ vyhlqd (26), α ≥ 0, ψC∈
∈ BV . Todi ρT g[ ] = ( [ ])α γ ψ− + , ρM g[ ] = ρ τ ψρ αT gg
T
[ ] [ ][ ]+ − −1 .
1. Linden C. N. On a conjecture of Valiron concerning sets of indirect Borel point // J. London Math.
Soc. – 1966. – 41. – P. 304 – 312.
2. DΩrbaßqn M. M. Yntehral\n¥e preobrazovanyq y predstavlenyq funkcyj v kompleksnoj
oblasty. – M.: Nauka, 1966. – 672 s.
3. Zyhmund A. Tryhonometryçeskye rqd¥: V 2 t. – M.: Myr, 1965. – T.C1, 2.
4. Chyzhykov I. E. Growth of harmonic functions in the unit disc and an application // Oberwolfach
Repts. – 2004. – 1, # 1. – P. 391 – 392.
5. Hardy G. H., Littlewood J. E. Some properties of fractional integrals. II // Math. Z. – 1931/32. –
34. – S. 403 – 439.
6. Íamoqn F. A. Neskol\ko zameçanyj k parametryçeskomu predstavlenyg klassov Nevan-
lynn¥ – DΩrbaßqna // Mat. zametky. – 1992. – 52, # 1. – S.C128 – 140.
7. Salem R. On a theorem of Zygmund // Duke Math. J. – 1943. – 10. – P. 23 – 31.
8. Uytteker ∏. T., Vatson DΩ. N. Kurs sovremennoho analyza: V 2 t. – M.: Fyzmathyz, 1963. –
T.C2. – 516 s.
9. Subxankulov M. A. Tauberov¥ teorem¥ s ostatkom. – M.: Nauka, 1976. – 400 s.
OderΩano 12.05.2005
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|
| id | umjimathkievua-article-3360 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:04Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4a/92c8371045aa2d5ec8a36ead8393854a.pdf |
| spelling | umjimathkievua-article-33602020-03-18T19:52:14Z On a complete description of the class of functions without zeros analytic in a disk and having given orders Про повний опис класу аналітичних у крузі функцій без нулів із заданими величинами порядків Chyzhykov., I. E. Чижиков, І. Е. For arbitrary $0 ≤ σ ≤ ρ ≤ σ + 1$, we describe the class $A_{σ}^{ρ}$ of functions $g(z)$ analytic in the unit disk $D = \{z : ∣z∣ < 1\}$ and such that $g(z) ≠ 0,\; ρ_T[g] = σ$, and $ρ_M[g] = ρ$, where $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$ $ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$. Для произвольных $0 ≤ σ ≤ ρ ≤ σ + 1$ описан класс $A_{σ}^{ρ}$ аналитических в единичном круге $D = \{z : ∣z∣ < 1\}$ функций $g(z)$ таких, что $g(z) ≠ 0,\; ρ_T[g] = σ$, $ρ_M[g] = ρ$, где $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$ $ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3360 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 979–995 Український математичний журнал; Том 59 № 7 (2007); 979–995 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3360/3464 https://umj.imath.kiev.ua/index.php/umj/article/view/3360/3465 Copyright (c) 2007 Chyzhykov. I. E. |
| spellingShingle | Chyzhykov., I. E. Чижиков, І. Е. On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title | On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title_alt | Про повний опис класу аналітичних у крузі функцій без нулів із заданими величинами порядків |
| title_full | On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title_fullStr | On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title_full_unstemmed | On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title_short | On a complete description of the class of functions without zeros analytic in a disk and having given orders |
| title_sort | on a complete description of the class of functions without zeros analytic in a disk and having given orders |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3360 |
| work_keys_str_mv | AT chyzhykovie onacompletedescriptionoftheclassoffunctionswithoutzerosanalyticinadiskandhavinggivenorders AT čižikovíe onacompletedescriptionoftheclassoffunctionswithoutzerosanalyticinadiskandhavinggivenorders AT chyzhykovie propovnijopisklasuanalítičnihukruzífunkcíjbeznulívízzadanimiveličinamiporâdkív AT čižikovíe propovnijopisklasuanalítičnihukruzífunkcíjbeznulívízzadanimiveličinamiporâdkív |