Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function
For entire functions whose zero counting functions are slowly increasing, we establish criteria for the regular growth of their logarithms of moduli and arguments in the metric of $L^p [0, 2π]$.
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| Дата: | 2010 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2922 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508920876892160 |
|---|---|
| author | Bodnar, O. V. Zabolotskii, N. V. Боднар, О. В.. Заболоцький, М. В. |
| author_facet | Bodnar, O. V. Zabolotskii, N. V. Боднар, О. В.. Заболоцький, М. В. |
| author_sort | Bodnar, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:27Z |
| description | For entire functions whose zero counting functions are slowly increasing, we establish criteria for the regular growth of their logarithms of moduli and arguments in the metric of $L^p [0, 2π]$. |
| first_indexed | 2026-03-24T02:32:53Z |
| format | Article |
| fulltext |
UDK 517. 53
O. V. Bodnar, M. V. Zaboloc\kyj (L\viv. nac. un-t im. I. Franka)
KRYTERI} REHULQRNOSTI ZROSTANNQ LOHARYFMA
MODULQ TA ARHUMENTU CILO} FUNKCI}
For entire functions whose counting function of zeros is slowly growing, we establish criteria of the
regular growth of their logarithm of modulus and argument in Lp 0 2, π[ ] -metric.
Dlq cel¥x funkcyj, sçetnaq funkcyq nulej kotor¥x qvlqetsq medlenno vozrastagwej, usta-
novlen¥ kryteryy rehulqrnoho rosta ee loharyfma modulq y arhumenta v Lp 0 2, π[ ] -metryke.
1. Vstup ta formulgvannq osnovnyx rezul\tativ. Dodatni, nespadni, neob-
meΩeni, neperervno dyferencijovni na R+ funkci] budemo nazyvaty funkciq-
my zrostannq. Funkci] zrostannq v i �v taki, wo v( )r ∼ �v( )r , r → + ∞, vva-
Ωatymemo ekvivalentnymy i budemo ototoΩngvaty. Klas funkcij zrostannq
v , dlq qkyx
r r
r
′v
v
( )
( )
→ 0 pry r → + ∞, poznaçymo çerez L. Vidomo [1, s. 15], wo
funkci] klasu L [ povil\no zrostagçymy i, navpaky, dlq dovil\no] povil\no
zrostagço] do + ∞ funkci] znajdet\sq funkciq z klasu L, ekvivalentna ]j.
Nexaj n r( ) = n r f( , , )0 — liçyl\na funkciq poslidovnosti nuliv ( )an cilo]
funkci] f, roztaßovanyx u porqdku nespadannq ]x moduliv. Poznaçymo çerez
H0( )L klas cilyx funkcij f nul\ovoho porqdku, nuli qkyx zadovol\nqgt\
umovu
∃ ∈v L ∃ >A 0 ∀ > <r n r f A r0 0: ( , , ) ( )v .
Ne zmenßugçy zahal\nosti, dali budemo vvaΩaty, wo f ( )0 1= , v( )0 0= .
V [2, s. 78] navedeno opys rehulqrnoho zrostannq loharyfma modulq, a v [3,
4] — arhumentu meromorfnyx funkcij dodatnoho porqdku v Lp 0 2, π[ ]-metryci
(dyv. takoΩ [5]). Metog dano] roboty [ vstanovlennq podibnyx rezul\tativ dlq
funkcij klasu H0( )L .
Dali vvaΩatymemo funkcig
ln ( )
( )
( )
f z
f w
f w
dw
z
= ′
∫
0
vyznaçenog v kompleksnij plowyni z radial\nymy rozrizamy vid nuliv cilo]
funkci] f do + ∞ . Funkciq ln ( )f z [ odnoznaçnog hilkog bahatoznaçno]
funkci] Ln f = ln ( )f z + i f zAng ( ) takog, wo ln ( )f 0 0= . Nexaj α j =
= ang a j , 0 ≤ α j < 2π, a j — nuli cilo] funkci] f. Qkwo poklasty α j
m( ) = α j +
+ 2πm , m ∈Z , to spivvidnoßennq
s r s r a( , ) ( , )ϕ − = 2 1π
α ϕa a rj j< ≤ ≤
∑
,
,
s r s r( , ) ( , )ϕ π ϕ+ −2 = s r a s r a( , ) ( , )+ −2 π
vyznaçagt\ dlq dovil\nyx çysel a ∈R i znaçen\ s r a( , ) sim’g mir s r( , )ϕ( ) na
odynyçnomu koli.
© O. V. BODNAR, M. V. ZABOLOC|KYJ, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7 885
886 O. V. BODNAR, M. V. ZABOLOC|KYJ
Poklademo dlq k ∈Z
n rk ( ) = e ik
a r
j
j
−
≤
∑ α , N r
n t
t
dtk
k
r
( )
( )
= ∫
0
.
Todi n r n r0( ) ( )= , N r N r0( ) ( )= i dlq koΩnoho ϕ0 ∈R pravyl\nymy [ riv-
nosti [2, s. 105]
n rk ( ) =
1
2
0
0 2
π
ϕϕ
ϕ
ϕ π
e ds rik−
+
∫ ( , ) .
Budemo hovoryty, wo nuli funkci] f L∈H0( ) magt\ kutovu wil\nist\, qk-
wo v usix toçkax neperervnosti ϕ i η deqko] miry µ na odynyçnomu koli isnu[
hranycq
lim
( , ) ( , )
( )r
s r s r
r→+∞
−ϕ η
v
= µ ϕ µ η( ) ( )− .
Z teoremy Karateodori – Levi (dyv., napryklad, [2, s. 98]) otrymu[mo, wo nuli
funkci] f iz klasu H0( )L magt\ kutovu wil\nist\ todi i lyße todi, koly dlq
dovil\noho k ∈Z isnugt\ skinçenni hranyci
δk
r
kn r
r
=
→+∞
lim
( )
( )v
. (1)
Teorema 1. Nexaj nuli funkci] f L∈H0( ) magt\ kutovu wil\nist\,
G i
k
ef
k
k
ik( )θ
δ θ=
≠
∑
0
.
Todi dlq dovil\noho p ∈ +∞[ )1, vykonu[t\sq
lim
arg ( )
( )
( )
/
r
i
f
p p
f re
r
G d
→+∞
−
=∫1
2
0
0
2
1
π
θ θ
θπ
v
, (2)
lim
ln ( )
( )
/
r
i p p
f re
r
d
→+∞
−
=∫1
2
00
0
2
1
π
δ θ
θπ
v1
, (3)
de v1( )r =
v( )t
t
dt
r
0∫ .
Teorema 2. Nexaj f naleΩyt\ H0( )L , dlq vsix k ∈Z \ 0{ } isnugt\ hra-
nyci (1) i funkciq G f ( )θ taka , qk v teoremi 1. Todi dlq dovil\noho p ∈
∈ 1, +∞[ )
lim
ln ( ) ( )
( )
( )
/
r
i
f
p p
f re N r
r
iG d
→+∞
− −
=∫1
2
0
0
2
1
π
θ θ
θπ
v
. (4)
Teorema 3. Nexaj f naleΩyt\ H0( )L i dlq deqkyx çysla p ∈ +∞[ )1, ta
funkci] H ∈ Lp 0 2, π[ ] vykonu[t\sq spivvidnoßennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
KRYTERI} REHULQRNOSTI ZROSTANNQ LOHARYFMA MODULQ … 887
lim
ln ( ) ( )
( )
( )
r
i p
f re N r
r
i H d
→+∞
−
−
∫
1
2 0
2
π
θ θ
θπ
v
=
1
0
/ p
.
Todi H d( )θ θ
π
0
2
∫ = 0 i dlq dovil\noho k ∈Z \ 0{ } isnugt\ hranyci
lim ( ) ( )
r
kN r r
→+∞
/v1 = ∆k .
Poznaçymo çerez L∗
pidklas funkcij zrostannq v klasu L takyj, wo
r r
r
′v
v
( )
( )
↘ 0 , r → + ∞.
Teorema 4. Nexaj f naleΩyt\ H0( )L∗
i dlq deqkyx çysel p ∈ +∞[ )1, ,
b0 ∈R ta funkci] H Lp∈ [ ]0 2, π vykonugt\sq umovy
lim
arg ( )
( )
( )
r
i p
f re
r
H d
→+∞
−
∫
1
2 0
2
π
θ θ
θπ
v
=
1
0
/ p
,
lim
ln ( )
( )r
i p
f re
r
b d
→+∞
−
∫
1
2
0
0
2
π
θ
θπ
v1
=
1
0
/ p
.
Todi nuli cilo] funkci] f magt\ kutovu wil\nist\, b0 0= δ , H G f( ) ( )θ θ=
dlq majΩe vsix θ π∈[ ]0 2, .
2. DopomiΩni tverdΩennq. Nexaj C rk ( , )Φ , k ∈Z , — koefici[nty Fur’[
funkci] Φ( )reiϕ
qk funkci] vid ϕ, tobto
C r re e dk
i ik( , ) ( )Φ Φ= −∫
1
2 0
2
π
ϕϕ ϕ
π
, r > 0.
Poklademo
l rk ( ) = l r fk ( , ) = C r fk ( , ln ) , c rk ( ) = c r fk ( , ) = C r fk , ln( ) ,
a rk ( ) = a r fk ( , ) = C r fk ( , arg ) .
Nexaj ln ( )f z = γ k
k
k
z
=
+∞∑ 1
— rozvynennq v deqkomu okoli toçky z = 0. Vraxo-
vugçy [2, s. 60], wo dlq cilo] funkci] porqdku ρ ≥ 0
γ k
j
k
jk a
= −
=
+∞
∑1 1
1
, k ≥ ρ[ ] + 1,
z formul dlq l rk ( ) , c rk ( ) , a rk ( ) (dyv., napryklad, [5; 2, s. 10; 3]) otrymu[mo
nastupni formuly dlq koefici[ntiv Fur’[.
Lema 1. Dlq cilo] funkci] f nul\ovoho porqdku spravdΩugt\sq rivnosti
l r r
n t
t
dtk
k k
k
r
( )
( )
= − +
+∞
∫ 1 , k ≥ 1, (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
888 O. V. BODNAR, M. V. ZABOLOC|KYJ
l r r
n t
t
dtk
k k
k
r
r
( )
( )
= +∫ 1 , k ≤ – 1, (6)
ia rk ( ) = − +
+∞
∫
1
2 1r
n t
t
dtk k
k
r
( )
–
1
2 1
0
r
n t
t
dtk k
k
r
−
− +∫
( )
, k ≥ 1, (7)
l r0( ) = c r0( ) = N r( ) , a r0( ) = 0, ia rk ( ) = ia rk− ( ) , k ≤ – 1. (8)
Dlq k ∈Z i funkci] γ, intehrovno] na 0, +∞[ ) , poklademo
Jk
k
k
r
r r
t
t
dt( , )
( )
γ
γ
= +
+∞
∫ 1 , Ik
k
k
r
r r
t
t
dt( , )
( )
γ
γ
= −
− +∫ 1
0
.
Lema 2. Dlq k ≥ 1 i v ∈ L spravdΩugt\sq spivvidnoßennq
Jk r( , )γ ∼ Ik r( , )γ ∼
v( )r
k
, r → + ∞.
Dovedennq. Za pravylom Lopitalq ma[mo
lim
( , )
( )r
k r
r→ +∞
J v
v
= lim
( )
( )r
k
r
k
t t dt
r r→ +∞
− −+∞
−
∫ v
v
1
= lim
( )
( ) ( )r
k
k k
r r
kr r r r→ +∞
− −
− − −
−
− + ′
v
v v
1
1 =
= lim
( )
( ) ( )r
r
k r r r→ +∞ − ′
v
v v
= lim
( )
( )
r
k
r r
r
→ +∞ − ′
1
v
v
=
1
k
i, analohiçno,
lim
( , )
( )r
k r
r→ +∞
I γ
v
=
v
v v
( )
( ) ( )
r r
kr r r r
k
k k
−
− + ′
1
1 = lim
( )
( )
r
k
r r
r
→ +∞ + ′
1
v
v
=
1
k
.
Lema 3. Nexaj f naleΩyt\ H0( )L . Todi
∃ >B 0 ∃ >r0 0 ∀ ∈ { }k Z\ 0 ∀ ≥r r0 :
l rk ( ) ≤
B
k
rv( ) , a rk ( ) ≤
B
k
rv( ) .
Dovedennq. Oskil\ky n rk ( ) ≤ n r( ) ≤ A rv( ) , to z formul (5) – (7) dlq
koefici[ntiv l rk ( ) i a rk ( ) oderΩu[mo
l rk ( ) ≤ r
n t
t
dtk
k
r
( )
+
+∞
∫ 1 ≤ A rkJ ( , )v , k ≥ 1,
l rk ( ) ≤ r
n t
t
dtk
k
r
( )
+∫ 1
0
≤ A rkI− ( , )v , k ≤ – 1,
a rk− ( ) = a rk ( ) ≤
1
2
A rkJ ( , )v +
1
2
A rkI ( , )v , k ≥ 1.
Dali, dlq k ≥ 1 i r > 0 ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
KRYTERI} REHULQRNOSTI ZROSTANNQ LOHARYFMA MODULQ … 889
Ik r( , )v = r
t
t
dtk
k
r
−
− +∫
v( )
1
0
≤
v( )r
r
t dt
k
k
r
−∫ 1
0
≤
v( )r
k
.
Oskil\ky v( )r — povil\no zrostagça funkciq pry r → + ∞, to v( )2 r ≤
3
2
v( )r
dlq r ≥ r0 . Tomu dlq k ≥ 1 i r ≥ r0 otrymu[mo
Jk r( , )v = r
t
t
dtk
k
r
v( )
+
+∞
∫ 1 = r
t
t
dtk
k
r
r
m m
m
v( )
+
=
+∞ +
∫∑ 1
2
2
0
1
≤ r r
kr
k m
k m
k
m
v( )2
21
0
+
−
=
+∞
∑ ≤
≤
2 v( ) ( )r
k
m
m
m
3
2
2
1
1
0
+
− +
=
+∞
∑ ≤
2 v( )r
k
m
m
3
4
1
0
+
=
+∞
∑ ≤
6 v( )r
k
,
wo dovodyt\ lemuO3.
Lema 4. Nexaj f naleΩyt\ H0( )L i vykonugt\sq spivvidnoßennq (1).
Todi
l rk ( ) ∼ ia rk ( ) ∼ −
δk
k
rv( ) , r → + ∞, k ∈ { }Z\ 0 .
Dovedennq. Z ohlqdu na lemuO1 l rk ( ) = − ( )Jk kr n r, ( ) dlq k ≥ 1, l rk ( ) =
= Ik kr n r, ( )( ) dlq k ≤ – 1, a rk ( ) = − ( )1
2
Jk kr n r, ( ) –
1
2
Ik kr n r, ( )( ) dlq k ≥ 1, a
za umovog lemy n rk ( ) ∼ δk rv( ) , r → + ∞. Tomu zhidno z lemog 2 otrymu[mo
(r → + ∞)
l rk ( ) ∼ −δk k rJ ( , )v ∼ −
δk
k
rv( ) , k ≥ 1,
l rk ( ) ∼ δk k rI− ( , )v ∼ −
δk
k
rv( ) , k ≤ – 1,
ia rk ( ) ∼ −
δk
k r
2
J ( , )v –
δk
k r
2
I ( , )v ∼ −
δk
k
rv( ) , k ≥ 1,
ia rk ( ) = − −ia rk ( ) ∼ −
−δ k
k
rv( ) = −
δk
k
rv( ) , k ≤ – 1,
oskil\ky δ−k = δk , wo dovodyt\ lemuO4.
Lema 5. Qkwo v naleΩyt\ L∗
, to funkciq ln ( )v1 r vhnuta vidnosno lo-
haryfma.
Dovedennq. Oskil\ky
v( )r =
t t
r
t
t
dt
r ′
∫
v
v
v( )
( )
( )
0
≥
r r
r
t
t
dt
r′
∫
v
v
v( )
( )
( )
0
=
r r
r
r
′v
v
v
( )
( )
( )1
i ′v1( )r r = v( )r , to
′v
v
( )
( )
r
r
≤
′v
v
1
1
( )
( )
r
r
.
Dali
d r
d r
2
1
2
ln ( )
ln
v
=
v
v
( )
( ) ln
r
r r1
′
=
r r
r
r
r
r
r
v
v
v
v
v
v
( )
( )
( )
( )
( )
( )1
1
1
′ − ′
≤ 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
890 O. V. BODNAR, M. V. ZABOLOC|KYJ
a otΩe, funkciq ln ( )v1 r vhnuta vidnosno loharyfma.
Z teorem 5 i 6 roboty [6] otrymu[mo nastupne tverdΩennq.
Lema 6. Nexaj funkciq g neperervno dyferencijovna, opukla na a, +∞[ ) ,
′g x( ) → + ∞ pry x → + ∞ i ln ( )g x — vhnuta funkciq. Todi dlq dovil\no]
opuklo] funkci] f z isnuvannq hranyci
lim
( )
( )x
f x
g x
K
→+∞
= , 0 < K < +∞,
vyplyva[ isnuvannq hranyci
lim
( )
( )x
f x
g x
K
→+∞
′
′
= .
3. Dovedennq teorem 1 i 2. Nexaj f naleΩyt\ H0( )L . Z isnuvannq dlq k ≠
≠ 0 hranyc\ (1) i ocinok
i
k
kδ
≤
A
k
vyplyva[, z uraxuvannqm teoremy Fißera –
Rissa, isnuvannq funkci] G f ( )θ i ]] naleΩnist\ do klasu L2 0 2, π[ ]. Z ohlqdu
na lemu 3 dlq r r≥ 0 ma[mo
l r
B
k
rk ( ) ( )≤ v , k ∈ { }Z\ 0 ,
a otΩe, poslidovnist\
l r
r k
k k( )
( )v
+
δ
naleΩyt\ prostoru lq pry vsix q > 1 i
r r> 0 . Todi, zastosuvavßy teoremu Xausdorfa – Gnha pry p ≥ 2 ,
1
p
+
1
q
= 1,
oderΩymo
1
2 0
2
π
θ θ
θπ
ln ( ) ( )
( )
( )
f re N r
r
i G d
i
f
p
−
−
∫ v
1/ p
≤
l r
r k
k k
k
q q
( )
( )
/
v
+
≠
∑ δ
0
1
.
Otrymanyj rqd zavdqky lemiO3 [ rivnomirno zbiΩnym na r0, +∞[ ) . Vykonav-
ßy u c\omu rqdi poçlennyj hranyçnyj perexid pry r → + ∞, z uraxuvannqm le-
myO4 otryma[mo spivvidnoßennq (4) dlq p ≥ 2. Zvidsy ta z nerivnosti Hel\dera
vstanovlg[mo tverdΩennq teoremy 2 i dlq 1 ≤ p < 2.
Iz spivvidnoßennq (4), vraxovugçy, wo dlq majΩe vsix θ ∈ 0 2, π[ ]
ln ( ) ( )
( )
( )
f re N r
r
i G
i
f
θ
θ
−
−
v
=
=
ln ( ) ( )
( )
f re N r
r
iθ −
v
+ i
f re
r
G
i
f
arg ( )
( )
( )
θ
θ
v
−
,
otrymu[mo (2) ta
lim
ln ( ) ( )
( )r
i p
f re N r
r
d
→+∞
−
∫
1
2 0
2
π
θ
θπ
v
1/ p
= 0.
Zavdqky pravylu Lopitalq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
KRYTERI} REHULQRNOSTI ZROSTANNQ LOHARYFMA MODULQ … 891
lim
( )
( )r
N r
r→+∞ v1
= lim
( )
( )r
n r
r→+∞ v
= δ0 , lim
( )
( )r
r
r→+∞
v
v1
= 0.
Tomu z ohlqdu na nerivnist\ trykutnyka dlq norm ta ostannix spivvidnoßen\
1
2 1
0
0
2
1
π
δ θ
θπ ln ( )
( )
/
f re
r
d
i p p
v
−
∫ ≤
1
2 10
2
1
π
θ
θπ ln ( ) ( )
( )
/
f re N r
r
d
i p p
−
∫ v
+
N r
r
( )
( )v1
0− δ → 0, r → + ∞,
wo dovodyt\ teoremuO1.
4. Dovedennq teorem 3 ta 4. Poznaçymo çerez dk koefici[nty Fur’[
funkci] H ( )θ . Todi dlq vsix k ∈N za umov teoremyO4 ma[mo
a r
r
dk
k
( )
( )v
− ≤
1
2 0
2
π
θ θ
θπ
arg ( )
( )
( )
f re
r
H d
i
v
−∫ ≤
≤
1
2 0
2
1
π
θ θ
θπ
arg ( )
( )
( )
/
f re
r
H d
i p p
v
−
∫ → 0, r → + ∞.
OtΩe, dlq vsix k ∈N
lim
( )
( )r
k
k
a r
r
d
→+∞
=
v
.
Dlq funkci] v ∈ L poklademo
v2( )r =
v1
0
( )t
t
dt
r
∫ =
dt
t
d
r t
0 0
∫ ∫
v( )τ
τ
τ .
Vraxovugçy, wo v( )r = o rv1( )( ) , v1( )r = o rv2( )( ) , r → + ∞, z obernenyx formul
[3] dlq koefici[ntiv Fur’[ arhumentu cilo] funkci]
N rk
∗( ) =
i
k
a rk ( ) – ik
dt
t
a
dk
tr
( )τ
τ
τ
00
∫∫ , de N r
N t
t
dtk
k
r
∗ = ∫( )
( )
0
,
otrymu[mo dlq koΩnoho k ∈ { }Z\ 0 isnuvannq hranyc\
lim
( )
( )r
kN r
r→+∞
∗
v2
= −
→+∞ ∫∫ik
r
dt
t
a
d
r
k
tr
lim
( )
( )1
2 00v
τ
τ
τ = −ik dk . (9)
Iz spivvidnoßen\
c r
r
b0
1
0
( )
( )v
− =
N r
r
b
( )
( )v1
0− ≤
ln ( )
( )
f re
r
b d
iθπ
θ
v1
0
0
2
−∫ ≤
≤
1
2 1
0
0
2
1
π
θ
θπ ln ( )
( )
/
f re
r
b d
i p p
v
−
∫ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
892 O. V. BODNAR, M. V. ZABOLOC|KYJ
umov teoremyO4 ta pravyla Lopitalq oderΩu[mo
lim
( )
( )r
N r
r→+∞
∗
0
2v
= lim
( )
( )r
N r
r→+∞ v1
= b0 . (10)
Poklademo
S r S r a( , ) ( , )ϕ − =
s t s t a
t
dt
r
( , ) ( , )ϕ −
∫
0
,
S r S r a∗ ∗−( , ) ( , )ϕ =
S r S r a
t
dt
r
( , ) ( , )ϕ −
∫
0
.
Todi [2, s. 105]
N rk ( ) =
1
2 0
2
π
ϕϕ
π
e dS rik−∫ ( , ) , N rk
∗( ) =
1
2 0
2
π
ϕϕ
π
e dS rik− ∗∫ ( , ) .
Z ohlqdu na (9), (10) ta teoremu Karateodori – Levi z ostannix rivnostej ot-
rymu[mo, wo poslidovnist\ mir ( )Sn
∗ = S rn
∗( ( , )ϕ / v2( )rn ) , de ( )rn — dovil\na
poslidovnist\ dodatnyx çysel, rn → +∞ , n → + ∞, [ zbiΩnog na odynyçnomu
koli, tobto isnu[ hranycq
lim
( , ) ( , )
( )r
S r S r
r→+∞
∗ ∗−ϕ η
v2
= µ ϕ µ η( ) ( )−
v usix toçkax neperervnosti ϕ i η deqko] miry µ na odynyçnomu koli.
Za lemogO5 funkciq ln ( )v1 r vhnuta vidnosno loharyfma, a tomu
r r
r
′v
v
1
1
( )
( )
=
=
v
v
( )
( )
r
r1
[ nezrostagçog na 0, +∞[ ) funkci[g. Zastosovugçy cg lemu do
funkci] v1( )r , perekonu[mos\, wo ln ( )v2 r [ vhnutog vidnosno loharyfma
funkci[g. Zastosovugçy dviçi lemuO6, oderΩu[mo isnuvannq hranyc\
lim
( , ) ( , )
( )r
S r S r
r→+∞
−ϕ η
v1
= lim
( , ) ( , )
( )r
s r s r
r→+∞
−ϕ η
v
= µ ϕ µ η( ) ( )− ,
wo dovodyt\ isnuvannq kutovo] wil\nosti poslidovnosti nuliv funkci] f ∈
∈ H0( )L∗
.
Znovu zastosuvavßy teoremu Karateodori – Levi, otryma[mo (dyv. (9), (10))
δ0 = lim
( )
( )r
n r
r→+∞
0
v
= b0 , δk = lim
( )
( )r
kn r
r→+∞ v
= −ik dk , k ≠ 0.
Zvidsy i z oznaçennq funkci] G f ( )θ vyplyva[, wo G f ( )θ = H ( )θ dlq majΩe
vsix θ π∈[ ]0 2, .
Teoremu 4 dovedeno.
Nexaj znovu dk — koefici[nty Fur’[ funkci] H ( )θ . Dlq k ≠ 0 ma[mo
C r f Nk ( , ln )− = l rk ( ) , C r f N0( , ln )− = 0 i, qk pry dovedenni teoremyO4,
oderΩu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
KRYTERI} REHULQRNOSTI ZROSTANNQ LOHARYFMA MODULQ … 893
l r
r
i dk
k
( )
( )v
− ≤
1
2 0
2
π
θ θ
θπ
ln ( ) ( )
( )
( )
f re N r
r
i H d
i p
−
−
∫ v
1/ p
→ 0, r → ∞.
Zvidsy d0 = H d( )θ θ
π
0
2
∫ = 0,
lim
( )
( )r
k
k
l r
r
i d
→+∞
=
v
, k ≠ 0. (11)
Z obernenyx formul [5, s. 981] dlq koefici[ntiv Fur’[ lk ma[mo
N r
r
k ( )
( )v1
=
l r
r
k ( )
( )v1
–
k
r
l t
t
dtk
r
v1 0( )
( )
∫ .
Zavdqky (11), spivvidnoßenng v( )r = o rv1( )( ) pry r → + ∞ ta pravylu Lopita-
lq z ostann\o] rivnosti otrymu[mo
lim
( )
( )r
k
k
N r
r
i k d
→+∞
= −
v1
, k ≠ 0,
wo dovodyt\ teoremuO3.
ZauvaΩymo, wo v [7] pobudovano pryklad cilo] funkci] f z klasu H0( )L ,
dlq qko] vykonu[t\sq umova teoremyO3 z H ( )θ ≡ 0, a hranycq lim
( )
( )r
N r
r→+∞
0
1v
ne
isnu[.
Avtory vdqçni profesorovi A.OA.OKondratgku, besidy z qkym spryqly poqvi
ci[] roboty.
1. Seneta E. Pravyl\no menqgwyesq funkcyy. – M.: Nauka, 1985. – 142 s.
2. Kondratgk A. A. Rqd¥ Fur\e y meromorfn¥e funkcyy. – L\vov: Vywa ßk., 1988. – 196 s.
3. Vasyl\kiv Q. V. Asymptotyçna povedinka loharyfmiçno] poxidno] ta loharyfmiv mero-
morfnyx funkcij cilkom rehulqrnoho zrostannq v Lp 0 2, π[ ] -metryci. 4.1 // Mat. stud. –
1999. – 12, # 1. – S. 37 – 58.
4. Vasyl\kiv Q. V. Asymptotyçna povedinka loharyfmiçno] poxidno] ta loharyfmiv mero-
morfnyx funkcij cilkom rehulqrnoho zrostannq v Lp 0 2, π[ ] -metryci. 4.2 // Tam Ωe. – # 2.
– S.O135 – 144.
5. Kalynec\ R. Z., Kondratgk A. A. Pro rehulqrnist\ zrostannq modulq i arhumentu cilo]
funkci] v L
p
0 2, π[ ] -metryci // Ukr. mat. Ωurn. – 1998. – 50, # 7. – S. 889 – 896.
6. Bratywev A. V. Ob obrawenyy pravyla Lopytalq // Mex. sploßnoj sred¥. – Rostov-na-Do-
nu: Yzd-vo Rostov. hos. un-ta, 1985. – S. 28 – 42.
7. Zabolotskii M. V. An example of entire function of strongly regular growth // Mat. Stud. – 2000. –
13, # 2. – P. 145 – 148.
OderΩano 02.02.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
|
| id | umjimathkievua-article-2922 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:32:53Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/dbe39793b4d5270f75d5745bf0e9bd12.pdf |
| spelling | umjimathkievua-article-29222020-03-18T19:40:27Z Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function Критерії регулярності зростання логарифма модуля та аргументу цілої функції Bodnar, O. V. Zabolotskii, N. V. Боднар, О. В.. Заболоцький, М. В. For entire functions whose zero counting functions are slowly increasing, we establish criteria for the regular growth of their logarithms of moduli and arguments in the metric of $L^p [0, 2π]$. Для целевых функций, счетная функция нулей которых является медленно возрастающей, установлены критерии регулярного роста ее логарифма модуля и аргумента в $L^p [0, 2π]$ -метрике. Institute of Mathematics, NAS of Ukraine 2010-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2922 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 7 (2010); 885–893 Український математичний журнал; Том 62 № 7 (2010); 885–893 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2922/2594 https://umj.imath.kiev.ua/index.php/umj/article/view/2922/2595 Copyright (c) 2010 Bodnar O. V.; Zabolotskii N. V. |
| spellingShingle | Bodnar, O. V. Zabolotskii, N. V. Боднар, О. В.. Заболоцький, М. В. Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title | Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title_alt | Критерії регулярності зростання логарифма модуля
та аргументу цілої функції |
| title_full | Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title_fullStr | Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title_full_unstemmed | Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title_short | Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| title_sort | criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2922 |
| work_keys_str_mv | AT bodnarov criteriafortheregularityofgrowthofthelogarithmofmodulusandtheargumentofanentirefunction AT zabolotskiinv criteriafortheregularityofgrowthofthelogarithmofmodulusandtheargumentofanentirefunction AT bodnarov criteriafortheregularityofgrowthofthelogarithmofmodulusandtheargumentofanentirefunction AT zabolocʹkijmv criteriafortheregularityofgrowthofthelogarithmofmodulusandtheargumentofanentirefunction AT bodnarov kriterííregulârnostízrostannâlogarifmamodulâtaargumentucíloífunkcíí AT zabolotskiinv kriterííregulârnostízrostannâlogarifmamodulâtaargumentucíloífunkcíí AT bodnarov kriterííregulârnostízrostannâlogarifmamodulâtaargumentucíloífunkcíí AT zabolocʹkijmv kriterííregulârnostízrostannâlogarifmamodulâtaargumentucíloífunkcíí |