On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point
The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.
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| Date: | 2005 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2005
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3617 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509736078671872 |
|---|---|
| author | Mokhonko, A. A. Мохонько, А. А. Мохонько, А. А. |
| author_facet | Mokhonko, A. A. Мохонько, А. А. Мохонько, А. А. |
| author_sort | Mokhonko, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity. |
| first_indexed | 2026-03-24T02:45:50Z |
| format | Article |
| fulltext |
UDK 517.923
A. A. Moxon\ko (Kyev. nac. un-t ym. T. Íevçenko)
O TEOREME MAL|MKVYSTA
DLQ REÍENYJ DYFFERENCYAL|NÁX URAVNENYJ
V�OKRESTNOSTY YZOLYROVANNOJ OSOBOJ TOÇKY
The statement of Malmquist’s theorem (1913) about the growth of meromorphic solutions of the
differential equation f ′ =
P z f
Q z f
( , )
( , )
, where P z f( , ) , Q z f( , ) are polynomials in all variables, is proved
for the case of solutions with isolated singularity at infinity.
TverdΩennq teoremy Mal\mkvista (1913) pro rist meromorfnyx rozv’qzkiv dyferencial\noho
rivnqnnq f ′ =
P z f
Q z f
( , )
( , )
, de P z f( , ) , Q z f( , ) — polinomy po vsix zminnyx, dovodyt\sq dlq vypadku
rozv’qzkiv z izol\ovanog osoblyvog toçkog v neskinçennosti.
Yspol\zuem oboznaçenyq teoryy meromorfn¥x funkcyj [1]. Symvol¥ Landau
o( )… , O( )… rassmatryvagtsq pry r → ∞ . Pust\ dano dyfferencyal\noe
uravnenye
f ′ =
P z f
Q z f
( , )
( , )
= j
t
j
j
j
s
j
j
p z f
p z f
=
=
∑
∑
0 1
0 2
( )
( )
, (1)
hde p zjq( ) — mnohoçlen¥. Esly v (1) deg f P ≤ 2, deg f Q = 0, to poluçaem
uravnenye Rykkaty f ′ = a z f2
2( ) + a z f1( ) + a z0( ), hde a zi( ) — racyonal\n¥e
funkcyy.
Yzvestna sledugwaq teorema Mal\mkvysta [2; 3, s. 67, 68]: esly uravnenye
(1) ne est\ uravnenye Rykkaty, to lgboe eho meromorfnoe reßenye qvlqetsq
racyonal\noj funkcyej. UtverΩdenye, πkvyvalentnoe teoreme Mal\mkvysta,
moΩno sformulyrovat\ v termynax nevanlynnovskyx xarakterystyk [4] (ysto-
ryg voprosa y byblyohrafyg sm. v [5, 6]): pust\ odnoznaçnaq meromorfnaq
funkcyq f z( ), z G∈ = { z : r0 ≤ z < + ∞ }, — reßenye dyfferencyal\noho
uravnenyq (1); esly (1) ne est\ uravnenye Rykkaty, to rost reßenyq ne prev¥-
ßaet rosta koπffycyentov:
T r f O T r p O r
j q
jq( , ) ( , ) (ln )
,
=
+∑ .
V nastoqwej stat\e πta teorema rasprostranqetsq na reßenyq s loharyfmy-
çeskoj osoboj toçkoj v ∞, a zatem na reßenyq s yzolyrovannoj osoboj toçkoj.
Uravnenyq pervoho porqdka, alhebrayçeskye otnosytel\no neyzvestnoj
funkcyy y ee proyzvodnoj, ne mohut ymet\ v yntehralax podvyΩn¥x trans-
cendentn¥x y suwestvenno osob¥x toçek [3, s.A54], odnako mohut ymet\ ne-
podvyΩn¥e transcendentn¥e y suwestvenno osob¥e toçky. Naprymer, yntehral
uravnenyq 2z f f ′ = 1 ymeet vyd f z( ) = ln( / )z C , C = const; funkcyq f z( ) =
= exp ln2 z( ) — reßenye uravnenyq z f ′ = 2 f zln .
Rassmotrym dyfferencyal\noe uravnenye (1), hde
p z h z z zjq jq
a bjq jq( ) ( ) (ln )= , h z c ojq jq( ) ( )= + 1 , cjq ∈C , ct1, cs2 0≠ , (2)
ajq , bjq ∈R , p zjq( ) , z G∈ = { z : r0 ≤ z < + ∞ }, — analytyçeskye funkcyy.
Budem predpolahat\, çto asymptotyçeskye sootnoßenyq (2) v¥polnqgtsq rav-
© A. A. MOXON|KO, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4 505
506 A. A. MOXON|KO
nomerno po θ v lgboj uhlovoj oblasty, a ymenno: ( ∀ α, β : – ∞ < α < β < + ∞ )
(∀ ε > 0 ) ( ∃ d = d ( α, β, ε ) > 0 ) : h zjq( ) = cjq + vjq z( ), vjq z( ) < ε, z ∈ { z = r eiθ
:
d ≤ r < + ∞, α ≤ θ ≤ β }, vjq z( ) — nekotoraq analytyçeskaq funkcyq.
Çerez Al oboznaçym mnoΩestvo analytyçeskyx v G = { z : r0 ≤ z < ∞ }
funkcyj, dlq kotor¥x ∞ qvlqetsq edynstvennoj osoboj toçkoj — loharyf-
myçeskoj osoboj toçkoj. MnoΩestvo Al qvlqetsq kommutatyvn¥m kol\com
bez delytelej nulq (celostn¥m kol\com). Çerez Ml oboznaçym pole çastn¥x
kol\ca Al (kaΩdoe celostnoe kol\co moΩno pohruzyt\ v nekotoroe pole [7
s.A52, 58]): Al ⊂ Ml . Esly f Al∈ , to budem hovoryt\, çto f z( ), z G∈ , —
funkcyq s yzolyrovannoj loharyfmyçeskoj osoboj toçkoj v ∞. Esly f Ml∈ ,
to funkcyq f z( ), z G∈ , naz¥vaetsq meromorfnoj funkcyej s loharyfmyçes-
koj osoboj toçkoj (nyΩe dano πkvyvalentnoe opredelenye meromorfnoj funk-
cyy s loharyfmyçeskoj osoboj toçkoj, osnovannoe na ponqtyy analytyçeskoho
prodolΩenyq).
Pust\ f z( ), z G∈ , — meromorfnaq funkcyq s loharyfmyçeskoj osoboj
toçkoj v ∞. V¥berem proyzvol\n¥e α, β; – ∞ < α < β < + ∞. PoloΩym k =
= π
β α−
. Rassmotrym uhlovug oblast\ gαβ = { z = r eiθ
: α ≤ θ ≤ β, 0 < r0 ≤ r <
< + ∞ } y sootvetstvugwug odnoznaçnug vetv\ f z( ), z g∈ αβ, funkcyy f z( ),
z G∈ . Nevanlynnovskye xarakterystyky vetvy f z( ), z g∈ αβ, opredelqgtsq
sledugwym obrazom [1, s.A40]:
A r f k
t
t
r
f t e f t e dt
r
r
k
k
k
i i
αβ
α β
π
( , ) ln ( ) ln ( )= −
+[ ]∫ +
−
+ +
0
1
1
1
2 ,
B r f k
r
f r e k dk
i
αβ
α
β
θ
π
θ α θ( , ) ln ( ) sin ( )= −( )∫ +2
, (3)
C r f k c t f
t
t
r
dt
r
r
k
k
kαβ αβ( , ) ( , )= +
∫ +
−
2 1
0
1
1
2 ,
hde
c t f c t k
r t
n
n n
αβ αβ
ρ α ψ β
ψ α( , ) ( , ) sin ( )
,
= ∞ = −( )
< ≤ ≤ ≤
∑
0
,
a ρ ψ
n
ie n
— polgs¥ funkcyy f z( ), z g∈ αβ, rassmatryvaem¥e s uçetom krat-
nosty,
S r f A r f B r f C r fαβ αβ αβ αβ( , ) ( , ) ( , ) ( , )= + + . (4)
V stat\e [8] dokazana sledugwaq teorema.
Teorema A. Pust\ meromorfnaq funkcyq f z( ), z G∈ , s loharyfmyçeskoj
osoboj toçkoj v ∞ f Ml∈( ) qvlqetsq reßenyem uravnenyq (1), koπffycyen-
t¥ pjq kotoroho opredelen¥ v (2). Esly (1) ne qvlqetsq uravnenyem Rykka-
ty f ′ = p z f21
2( ) + p z f11( ) + p z01( ) , to rost reßenyq ne prev¥ßaet rosta
koπffycyentov, t.,e. dlq lgboj vetvy f z( ), z g∈ αβ, v¥polnqetsq
S r f O S r p O O
j q
jqαβ αβ( , ) ( , ) ( ) ( )
,
=
+ =∑ 1 1 . (5)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O TEOREME MAL|MKVYSTA DLQ REÍENYJ … 507
Teoremu A moΩno utoçnyt\, esly rassmatryvat\ reßenyq, prynadleΩawye
kol\cu Al , Al ⊂ Ml . A ymenno, budet dokazana takaq teorema.
Teorema 1. Pust\ funkcyq f z( ), z G∈ , s yzolyrovannoj loharyfmyçeskoj
osoboj toçkoj v ∞ ( f Al∈ , Al ⊂ Ml ) qvlqetsq reßenyem uravnenyq (1), (2).
Esly (1) ne qvlqetsq lynejn¥m uravnenyem vyda f ′ = p z f11( ) + p z01( ) , to dlq
lgboj vetvy f z( ), z g∈ αβ, v¥polnqetsq sootnoßenye (5).
Analohyçnoe svojstvo ymegt y reßenyq, ymegwye yzolyrovannug osobug
toçku lgboj pryrod¥ (suwestvenno osobug, alhebrayçeskug, loharyfmyçes-
kug, polgs).
Napomnym opredelenye nevanlynnovskyx xarakterystyk odnoznaçnoj mero-
morfnoj funkcyy f z( ), z G∈ = { z : r0 ≤ z < + ∞ }. Çerez n r f( , ) oboznaçym
çyslo polgsov funkcyy f v kol\ce { z : r0 ≤ z ≤ r }. Dlq x ≥ 0 oboznaçym
ln+ x = max ln ,x 0( ). Tohda [1, s.A23]
m r f f r e di( , ) ln= ( )∫ +1
2
0
2
π
ϕ
π
ϕ
,
(6)
N r f
n t f
t
dt
r
r
( , )
( , )= ∫
0
, T r f m r f N r f( , ) ( , ) ( , )= + .
Analohyçno opredelqgtsq nevanlynnovskye xarakterystyky m r f( , ), N r f( , ) ,
T r f( , ) dlq ν-znaçn¥x funkcyj f z( ), z G∈ , ymegwyx v ∞ alhebrayçeskug
toçku vetvlenyq (sm. [9]).
Teorema 2. Pust\ funkcyq f z( ), z G∈ , s yzolyrovannoj osoboj toçkoj v
∞ qvlqetsq reßenyem uravnenyq (1), (2). Esly (1) ne qvlqetsq lynejn¥m
uravnenyem, to rost reßenyq ne prev¥ßaet rosta koπffycyentov, t.,e. lybo
dlq lgboj vetvy f z( ), z g∈ αβ, v¥polnqetsq sootnoßenye (5), lybo (esly
f z( ), z G∈ , — odnoznaçnaq holomorfnaq yly ν -znaçnaq alhebroydnaq funk-
cyq) v¥polnqetsq sootnoßenye
T r f O T r p O r O r
j q
jq( , ) ( , ) (ln ) (ln )
,
=
+ =∑ , r → + ∞ . (7)
Utoçnym, kak m¥ ponymaem operacyy nad mnohoznaçn¥my funkcyqmy. Ras-
smotrym kruh g = { z : z r− 0 < ε }, hde r0, ε > 0 ( ε — dostatoçno maloe). V¥-
berem kakye-nybud\ pravyl\n¥e πlement¥ [10, s.A480] exp lna zjq 0( ) , ln0 z bjq( ) ,
z g∈ , sootvetstvenno funkcyj z
a jq = exp lna zjq( ) , ln z bjq( ) . Yz svojstv πtyx
funkcyj sleduet, çto v¥brann¥e πlement¥ moΩno analytyçesky prodolΩyt\
vdol\ lgboj neprer¥vnoj kryvoj v oblasty G = { z : r0 ≤ z < + ∞ }. Pred-
poloΩym, çto suwestvuet pravyl\n¥j πlement f z0( ) , z g∈ , takoj, çto pry
podstanovke f z0( ) , exp lna zjq 0( ) , ln0 z bjq( ) , z g∈ , v (1), (2) vmesto sootvet-
stvenno f, z
a jq , ln z bjq( ) poluçaem toΩdestvo pry z g∈ . M¥ predpolahaem,
çto πlement f z0( ) , z g∈ , moΩno analytyçesky prodolΩyt\ vdol\ lgboj
neprer¥vnoj kryvoj z = λ( )t , t0 ≤ t ≤ t1, λ( )t0 = r0
, λ( )t1 = z1
, prynadleΩa-
wej G, pryçem rezul\tatom prodolΩenyq qvlqetsq lybo pravyl\n¥j πlement
f z1( ), z ∈ { z : z z− 1 < ε1}, ε1 > 0, lybo πlement, ymegwyj v toçke z1 neraz-
vetvlenn¥j polgs (πlement vyda
j s j
ja z z= −
+∞∑ −( )1 , s ∈N). PredpoloΩym, çto
dlq lgboho z G1 ∈ suwestvuet beskoneçnoe mnoΩestvo razlyçn¥x πlementov
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
508 A. A. MOXON|KO
ukazannoho vyda s centrom z1, kotor¥e qvlqgtsq neposredstvenn¥my
analytyçeskymy prodolΩenyqmy πlementa f z0( ) , z g∈ . MnoΩestvo vsex takyx
πlementov oboznaçym çerez f z( ), z G∈ . Budem hovoryt\, çto f z( ), z G∈ , —
meromorfnaq funkcyq s loharyfmyçeskoj osoboj toçkoj v ∞, f Ml∈ . V
çastnosty, esly pry vsex analytyçeskyx prodolΩenyqx πlementa f z0( ) , z g∈ ,
v oblasty G rezul\tatom prodolΩenyq qvlqetsq pravyl\n¥j πlement, to
f z( ), z G∈ , ymeet v ∞ yzolyrovannug loharyfmyçeskug osobug toçku
f Al∈( ).
V¥berem proyzvol\n¥e α, β; – ∞ < α < β < + ∞. Pust\, naprymer, α > 0.
Rassmotrym kryvug z = r eit
0 = µ( )t , 0 ≤ t ≤ α, µ( )0 = r0
, µ α( ) = r ei
0
α
.
Analytyçesky prodolΩym πlement¥ f z0( ) , exp lna zjq 0( ) , ln0 z bjq( ) , z g∈ ,
vdol\ kryvoj µ( )t , 0 ≤ t ≤ α. V rezul\tate prodolΩenyq poluçym πlement¥
f zα( ) , exp lna zjq α( ) , lnα z bjq( ) s centrom v toçke r ei
0
α
. Dalee analytyçesky
prodolΩym πty πlement¥ vdol\ vsevozmoΩn¥x kryv¥x z = r t ei t( ) ( )θ , t ∈ t t1 2,[ ],
hde r t( ), θ( )t , t1 ≤ t ≤ t2 , — neprer¥vn¥e funkcyy, takye, çto r0 ≤ r t( ) < + ∞,
α ≤ θ( )t ≤ β , t1 ≤ t ≤ t2 . MnoΩestvo vsex πlementov, poluçenn¥x v rezul\tate
takyx prodolΩenyj, budem oboznaçat\ sootvetstvenno çerez
f z( ), z g∈ αβ = z r e r ri= ≤ ≤ ≤ < + ∞{ }θ α θ β: , 0 ,
(8)
z
a jq
, z g∈ αβ, ln z bjq( ) , z g∈ αβ;
gαβ — uhlovaq oblast\ na rymanovoj poverxnosty funkcyy f z( ), z G∈ . Esly
β – α < 2π, to sohlasno teoreme o monodromyy [10, s.A488] funkcyy (8) —
odnoznaçn¥e analytyçeskye funkcyy v oblasty gαβ ⊂ C. Esly β – α ≥ 2π, to
oblast\ gαβ moΩno rassmatryvat\ kak odnosvqznug oblast\ na rymanovoj po-
verxnosty funkcyy f z( ), z G∈ . V πtoj oblasty takΩe prymenyma teorema o
monodromyy. Poπtomu funkcyy (8) — odnoznaçn¥e analytyçeskye funkcyy na
kuske rymanovoj poverxnosty gαβ .
Spravedlyva sledugwaq teorema [11]: pust\
F =
P f
Q f
( )
( )
= j
t
j
j
j
s
j
j
p f
p f
=
=
∑
∑
0 1
0 2
, d t s= max( , ) ,
f, p Mjq l∈ , pt1, ps2 ≠ 0, pryçem P f( ) , Q f( ) vzaymno prost¥, kak mnohoçlen¥
ot f nad polem Ml . Tohda
S r F dS r f O S r p O
j q
jqαβ αβ αβ( , ) ( , ) ( , ) ( )
,
= +
+∑ 1 . (9)
Esly f, p Mjq ∈ , M — pole odnoznaçn¥x meromorfn¥x yly alhebroydn¥x v ob-
lasty G funkcyj, pryçem P f( ) , Q f( ) vzaymno prost¥, kak mnohoçlen¥ ot f
nad polem M, to
T r F dT r f O T r p O r
j q
jq( , ) ( , ) ( , ) (ln )
,
= +
+∑ . (10)
Nam ponadobytsq sledugwaq lemma (sm. [8], formula (14)).
Lemma 1. Pust\ f z( ), z ∈ { z = r eiθ
: α1 ≤ θ ≤ β1, r0 ≤ r < + ∞ }, — mero-
morfnaq funkcyq. Esly α1 ≤ α < β ≤ β1, to
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O TEOREME MAL|MKVYSTA DLQ REÍENYJ … 509
S r f S r f Oα β αβ1 1
1( , ) ( , ) ( )≥ + . (11)
Dokazatel\stvo teorem¥ 1. Kak sleduet yz teorem¥ A, esly uravnenye
(1), (2) ymeet reßenye f ∈ Al ⊂ M y uravnenye (1) ne qvlqetsq uravnenyem Ryk-
katy (a sledovatel\no, y lynejn¥m uravnenyem), to dlq lgboj vetvy f z( ),
z g∈ αβ, v¥polnqetsq sootnoßenye (5).
Pust\ teper\ (1) — uravnenye Rykkaty, t.Ae. ymeet vyd
′ = + +f p z f p z f p z21
2
11 01( ) ( ) ( ). (12)
PokaΩem, çto esly v (12) koπffycyent p z21( ) � 0, to takΩe v¥polnqetsq so-
otnoßenye (5).
Prymenqq k (12) formulu (9), poluçaem
S r f S r f O S r p O
j
jαβ αβ αβ( , ) ( , ) ( , ) ( )′ = +
+
=
∑2 1
0
2
1 . (13)
Yzvestno [12] (teorema 1), çto meromorfnoe reßenye f z( ), z G∈ , s loha-
ryfmyçeskoj osoboj toçkoj v ∞ dyfferencyal\noho uravnenyq (1) s koπf-
fycyentamy p zjq( ) vyda (2) ymeet koneçn¥j porqdok rosta p.
Pust\ A, B takye, çto A < α < β < B. Rassmotrym odnoznaçn¥e vetvy f z( ),
z g∈ αβ y f z( ), z gAB∈ = { z = r eiθ
: A ≤ θ ≤ B, r0 ≤ r < + ∞ } funkcyy f z( ),
z G∈ . Pust\ { }cq — mnoΩestvo vsex nulej y polgsov vetvy f z( ), z gAB∈ .
V¥berem proyzvol\noe ε > 0 y dlq kaΩdoho c cq q∈{ } postroym okruΩnost\ s
centrom cq radyusa δq = cq
p− − −1 2ε /
. Çerez E oboznaçym mnoΩestvo toçek
oblasty gAB rymanovoj poverxnosty funkcyy f z( ), z G∈ , leΩawyx vnutry
vsex πtyx okruΩnostej. Tohda [12] (lemma 4) (∃ d = d A B( , , )ε > 0) :
′ < + +f z
f z
z p( )
( )
2 2 ε
, z g EAB∈ \ , z d≥ ,
(14)
∑ ∑= < = < +∞− − −δ ε
q q
p
c K
1 2/
const , c cq q∈{ }.
Dlq kaΩdoho c cq q∈{ } postroym ynterval cq q−[ δ , cq q+ ]δ . Pust\ ∆ —
mnoΩestvo toçek, prynadleΩawyx πtym yntervalam. Yz (14) sleduet, çto E —
mnoΩestvo kruhov s koneçnoj summoj radyusov, mes ∆ < 2K.
MoΩno sçytat\, çto luçy Λ( )α = { z = r eiα
: r ≥ d }, Λ( )β = { z = r eiβ
: r ≥ d }
ne peresekagtsq s E, kohda d — dostatoçno bol\ßoe (E ∩ ( Λ( )α ∪ Λ( )β ) =
= ∅). Dejstvytel\no, poskol\ku E — mnoΩestvo kruhov s koneçnoj summoj
radyusov, to (∃ α1: A < α1 < α ) ( ∃ d = d A( , )α > 0 ) takoe, çto luç Λ( )α1 = { z :
z = r eiα1
: r ≥ d } ne peresekaet kruhy yz mnoΩestva E, ( Λ( )α1 ∩ E = ∅ ) [13]
(formula (31)). Analohyçno suwestvuet β1, β < β1 < B, takoe, çto luç Λ( )β1 =
= { z = r eiβ1
: r ≥ d } ne peresekaet kruhy yz E, ( Λ( )β1 ∩ E = ∅ ). Poπtomu
vmesto vetvy f z( ), z g∈ αβ, moΩno rassmatryvat\ vetv\ f z( ), z g∈ α β1 1
, hde A <
< α1 ≤ α < β ≤ β1 < B.
Esly r ∉∆ , to, uçyt¥vaq (14) y to, çto k = π
β α−
> 0, sin ( )k θ α−( ) ≥ 0, α ≤
≤ θ ≤ β, poluçaem
′( )
( )
f r e
f r e
i
i
θ
θ < r p2 2+ + ε , α ≤ θ ≤ β,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
510 A. A. MOXON|KO
B r
f
fαβ , ′
= 2k
r
f r e
f r e
k dk
i
iπ
θ α θ
α
β θ
θ∫ + ′( )
( ) −ln sin ( ) <
< 2 2 21k p r rkπ ε β α− −+ + −( ) ln ( ) = o( )1 , r → + ∞, r ∉∆ . (15)
Na luçax Λ( )α , Λ( )β v¥polnqetsq ocenka (14). Poπtomu
A r
f
fαβ , ′
= k
t
t
r
f t e
f t e
f t e
f t e
dt
t
r
r
k
k
k
i
i
i
iπ
α
α
β
β
0
1
2∫ −
′( )
( ) +
′( )
( )
+ +ln ln =
= k
r
d
d
r
π
0
∫ ∫… + …
< O( )1 + 2 1 2 2
2
k
t
t
r
p t dt
t
d
r
k
k
kπ
ε∫ −
+ +( ) ln
= O( )1 . (16)
Dalee, v¥polnqgtsq ocenky [1, s.A45] (formula (6.9))
B r fαβ , ′( ) = B r f
f
fαβ , ′
≤ B r fαβ ,( ) + B r f
f
fαβ , ′
,
A r fαβ , ′( ) = A r f
f
fαβ , ′
≤ A r fαβ ,( ) + A r f
f
fαβ , ′
.
Poπtomu, uçyt¥vaq (15), (16), poluçaem
B r fαβ , ′( ) ≤ B r fαβ ,( ) + o( )1 , r ∉∆ , r → + ∞,
(17)
A r fαβ , ′( ) ≤ A r fαβ ,( ) + O( )1 .
Poskol\ku funkcyq f z( ), z G∈ , s yzolyrovannoj loharyfmyçeskoj osoboj
toçkoj ne ymeet polgsov, to (sm. (3))
C r fαβ ,( ) = C r fαβ , ′( ) ≡ 0, r ≥ r0
. (18)
Yz (18), (17), (4) sleduet
S r fαβ , ′( ) ≤ S r fαβ ,( ) + O( )1 , r ∉∆ . (19)
Uçyt¥vaq (13), (19), ymeem
S r fαβ ,( ) ≥ 2 S r fαβ ,( ) + O S r p
j
j
=
∑ ( )
0
2
1αβ , + O( )1 ,
(20)
S r fαβ ,( ) = O S r p
j
j
=
∑ ( )
0
2
1αβ , + O( )1 , r ∉∆ , mes ∆ < + ∞.
Yz (2) – (4) sleduet
S r pjqαβ ,( ) = O( )1 . (21)
Otsgda s uçetom (20) poluçaem
S r fαβ ,( ) = O( )1 , r ∉∆ . (22)
PokaΩem, çto v (22) ysklgçytel\noe mnoΩestvo ∆ moΩno opustyt\. Suwest-
vuet neub¥vagwaq neprer¥vnaq funkcyq
�
S r fαβ ,( ) takaq, çto [1, s.A43]
�
S r fαβ ,( ) = S r fαβ ,( ) + O( )1 . (23)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O TEOREME MAL|MKVYSTA DLQ REÍENYJ … 511
Yz (22), (23) poluçaem
�
S r fαβ ,( ) = O( )1 , r ∉∆ , mes ∆ < ∞. Poskol\ku
�
S r fαβ ,( )
— neub¥vagwaq funkcyq, yz pred¥duweho sleduet
�
S r fαβ ,( ) < C = const ∀ r ≥
≥ r0
. Poπtomu s uçetom (23)
S r fαβ ,( ) < const ∀ ≥r r0. (24)
Otsgda y yz (21) sleduet sootnoßenye (5).
Esly b¥ luç Λ( )α = { z = r eiα
: r ≥ d } yly luç Λ( )β = { z = r eiβ
: r ≥ d } pry
lgbom d peresekal mnoΩestvo E (sm. (14)), to, kak otmeçalos\ v¥ße, m¥
rassmatryvaly b¥ vetv\ f z( ), z g∈ α β1 1
, A < α1 < α < β < β1 < B, y analohyçno
pred¥duwemu dokazaly b¥ ocenku S r fα β1 1
,( ) = O( )1 . Poskol\ku α1 < α < β <
< β1, yz (11) sleduet S r fαβ ,( ) < S r fα β1 1
,( ) + O( )1 = O( )1 . Poπtomu ocenka (5)
spravedlyva dlq lgboj vetvy.
Ostalos\ rassmotret\ sluçaj, kohda v (12) p z21 0( ) ≡ . V πtom sluçae (12),
(1)Aqvlqetsq lynejn¥m uravnenyem f ′ = p z f11( ) + p z01( ) , hde p zj1( ) opredele-
n¥ vA(2).
Teorema 1 dokazana.
Dokazatel\stvo teorem¥ 2. Dlq funkcyy f z( ), z G∈ , s yzolyrovannoj
osoboj toçkoj v ∞ vozmoΩn¥ try predpoloΩenyq: 1) funkcyq ymeet v ∞
loharyfmyçeskug osobug toçku (πtot sluçaj rassmotren v teoreme 1); 2) f z( ),
z G∈ , — odnoznaçnaq holomorfnaq funkcyq; 3) f z( ), z G∈ , qvlqetsq ν-
znaçnoj alhebroydnoj funkcyej, pryçem f z( ) =
n n
nz= −∞
+∞∑ α ν/ , z G∈ , ν > 1,
ν ∈N . Pust\ teper\ reßenyem uravnenyq (1), (2) qvlqetsq lybo odnoznaç-
naqAAholomorfnaq funkcyq f z( ), z G∈ , lybo ν-znaçnaq funkcyq f z( ) =
=
n n
nz= −∞
+∞∑ α ν/ , z G∈ .
V [3, s.A67] yz-za sloΩnosty dokazatel\stva pryvodytsq tol\ko formulyrov-
ka teorem¥ Mal\mkvysta [2]. Prostoe dokazatel\stvo teorem¥ moΩno polu-
çyt\ metodom Josyd¥ [4], yspol\zuq formulu (10). V¥polnym v (1) zamenu f =
= u−1 + κ, hde κ — takaq konstanta, çto P z( , )κ � 0, Q z( , )κ � 0. V rezul\ta-
te poluçym
′ =u
R z u
V z u
( , )
( , )
, (25)
hde R, V — mnohoçlen¥ otnosytel\no u s koπffycyentamy P zjq( ) vyda (2),
qvlqgwymysq lynejn¥my kombynacyqmy koπffycyentov p zjq( ) uravnenyq
(1), (2). Stepeny R, V otnosytel\no u sootvetstvenno ravn¥ t y t – 2 (esly
t – 2 ≥ s) y s + 2 y s (esly t – 2 < s ). Pust\, dlq opredelennosty, t – 2 < s.
Tohda deg /u R V = s + 2. Prymenqq k (25) formulu (10), poluçaem
T r u s T r u O T r P O rjq( , ) ( ) ( , ) ( , ) (ln )′ = + + ( ) +∑2 . (26)
Poskol\ku koπffycyent¥ P zjq( ) qvlqgtsq lynejn¥my kombynacyqmy ko-
πffycyentov p zjq( ) uravnenyq (1), (2), yz svojstv xarakterystyky T r f( , )
sleduet [1, s.A45] (formul¥ (6.5) – (6.7)) T r Pjq( , ) = O T r pjq( , )∑( ) + O( )1 . Ot-
sgda s uçetom (26) ymeem
T r u s T r u O T r p O rjq( , ) ( ) ( , ) ( , ) (ln )′ = + + ( ) +∑2 . (27)
Tak kak funkcyq u z( ) , z G∈ , ne ymeet toçek vetvlenyq, otlyçn¥x ot ∞ ,
polgs¥ funkcyj ′u z( ), z G∈ , y u z( ) , z G∈ , raspoloΩen¥ v odnyx y tex Ωe
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
512 A. A. MOXON|KO
toçkax. KaΩdomu polgsu porqdka m funkcyy u z( ) sootvetstvuet polgs po-
rqdka m + 1 proyzvodnoj ′u z( ). Poπtomu n r u( , )′ ≤ 2n r u( , ) [1, s.A131],
N r u N r u( , ) ( , )′ ≤ 2 . (28)
Dlq odnoznaçnoj meromorfnoj funkcyy u z( ) , z G∈ , y dlq ν-znaçnoj
funkcyy u z( ) , z G∈ , spravedlyva lemma o loharyfmyçeskoj proyzvodnoj [1,
s.A122] (teorema 1.3), [9]:
m r u
u
o T r u, ( , )′
= ( ), r ∉∆ , mes ∆ < ∞,
poπtomu [1, s.A44] (formula (6.1))
m r u( , )′ = m r u u
u
, ′
≤ m r u( , ) + m r u
u
, ′
= m r u( , ) + o T r u( , )( ), r ∉∆ .
Otsgda, uçyt¥vaq (28), poluçaem
T r u( , )′ = N r u( , )′ + m r u( , )′ ≤ 2N r u( , ) + m r u( , ) + o T r u( , )( ) ≤
≤ 2 1+( )o T r u( ) ( , ), r ∉∆ , mes ∆ < ∞. (29)
Yz (27), (29) sleduet
2 1+( )o T r u( ) ( , ) ≥ ( ) ( , )s T r u+ 2 + O T r pjq( , )∑( ) + O r(ln ), r ∉∆ ,
(30)
s o T r u+( )( ) ( , )1 = O T r pjq( , )∑( ) + O r(ln ), r ∉∆ .
Esly s > 0, a znaçyt, uravnenye (1) ne qvlqetsq uravnenyem Rykkaty (a tem bo-
lee lynejn¥m uravnenyem), to (30) moΩno zapysat\ sledugwym obrazom:
T r u( , ) = O T r pjq( , )∑( ) + O r(ln ), r ∉∆ . (31)
Uçyt¥vaq (2), (6), poluçaem
T r pjq( , ) = O r(ln ) . (32)
Yz (31), (32) sleduet, çto suwestvuet M = const > 0 takoe, çto
T r u( , ) < M ln r, r ∉∆ , mes ∆ < K = const. (33)
Pust\ r > K. Poskol\ku mes ∆ < K , to ∃ ∈ +[ ]r r r K1 , , r1 ∉∆ . Funkcyq
T r u( , ), r ≥ r0
, — vozrastagwaq [1, s.A33] (teorema 4.3), poπtomu, uçyt¥vaq (33),
ymeem
T r u( , ) < T r u( , )1 < M rln 1 < M rln( )2 < 2 M rln ∀ ≥ ( )r Kmax , ln 2 .
Sledovatel\no,
T r u( , ) = O r(ln ), r ≥ r0
. (34)
Poskol\ku u = 1
f − κ
, κ = const, to, prymenqq pervug osnovnug teoremu
Nevanlynn¥ [1, s.A27] (teorema 4.1), poluçaem T r u( , ) = T r
f
, 1
−
κ
= T r f( , ) +
+ O( )1 . Otsgda y yz (31), (34) sleduet (7).
Pust\ s = 0. Po predpoloΩenyg t – 2 < s. Sledovatel\no, 0 ≤ t < s + 2 = 2,
poπtomu s = 0, t = 1. Takym obrazom, uravnenye (1) qvlqetsq lynejn¥m uravne-
nyem f ′ = p z f11( ) + p z01( ) , hde p zj1( ) opredelen¥ v (2). Analohyçno yssledu-
etsq sluçaj t – 2 ≥ s.
Teorema 2 dokazana.
Zameçanye. Pust\ f z( ) — analytyçeskaq funkcyq s yzolyrovannoj su-
westvenno osoboj toçkoj v ∞ (naprymer, celaq transcendentnaq funkcyq)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O TEOREME MAL|MKVYSTA DLQ REÍENYJ … 513
yly ν-znaçnaq analytyçeskaq funkcyq s alhebrayçeskoj toçkoj vetvlenyq v
∞. Zapyßem ee arhument v pokazatel\noj forme; funkcyq f r eiθ( ), r 0
≤ r <
< + ∞, – ∞ < θ < + ∞, ymeet po θ peryod 2π (sootvetstvenno, peryod 2πν ). ∏to
pozvolqet rassmatryvat\ funkcyg f r eiθ( ) s suwestvenno osoboj toçkoj (s al-
hebrayçeskoj toçkoj vetvlenyq) v ∞ kak raznovydnost\ funkcyy s loharyf-
myçeskoj osoboj toçkoj v ∞, ymegwej po θ peryod 2π (sootvetstvenno, pe-
ryod 2πν ) (sm. v¥ße opredelenye funkcyy f Ml∈ , osnovannoe na ponqtyy
analytyçeskoho prodolΩenyq). Poπtomu ocenka (5) rosta reßenyq s loharyf-
myçeskoj osoboj toçkoj prymenyma takΩe k reßenyqm s suwestvenno osoboj
yly alhebrayçeskoj toçkoj vetvlenyq. Dostatoçno v (5) vzqt\ α = 0, β = 2π y
rassmatryvat\ xarakterystyku S r f0 2, ( , )π .
Esly Ωe ocenyvat\ rost reßenyq s pomow\g xarakterystyky T r f( , ) , to
nuΩno dopolnytel\no predpoloΩyt\, çto koπffycyent¥ p zjq( ) (sm. (2)) pry-
nadleΩat polg funkcyj, v kotorom prymenyma formula (10): neobxodymo sçy-
tat\, çto v (2) pokazately stepenej ajq ∈Q , bjq = 0 .
1. Hol\dberh A. A., Ostrovskyj Y. V. Raspredelenye znaçenyj meromorfn¥x funkcyj. – M.:
Nauka, 1970. – 592 s.
2. Malmquist J. Sur les fonctions á un nombre fini de branches défínes par les équations différentielles
du premier order // Acta Math. – 1913. – 36. – P. 297 – 343.
3. Holubev V. V. Lekcyy po analytyçeskoj teoryy dyfferencyal\n¥x uravnenyj. – M.; L.:
Hostexteoryzdat, 1950. – 436 s.
4. Yosida K. A generalization of a Malmquist’s theorem // Jap. J. Math. – 1933. – 9. – P. 253 – 256.
5. Hol\dberh A. A., Levyn B. Q., Ostrovskyj Y. V. Cel¥e y meromorfn¥e funkcyy // Ytohy
nauky y texnyky. Sovr. probl. matematyky. Fundam. napravlenyq / VYNYTY. – 1991. – 85. –
S. 5 – 186.
6. Laine I. Nevanlinna theory and complex differential equations. – Berlin; New York: Walter
Gruyter, 1993. – 400 p.
7. Van der Varden B. L. Alhebra. – M.: Nauka, 1979. – 624 s.
8. Moxon\ko A. A. Teorema Mal\mkvysta dlq reßenyj dyfferencyal\n¥x uravnenyj
vAAokrestnosty loharyfmyçeskoj osoboj toçky // Ukr. mat. Ωurn. – 2004. – 56, # 4. –
S.A476A–A483.
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# 1–2. – P. 17 – 39.
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// Syb. mat. Ωurn. – 1981. – 22, # 3. – S. 214 – 218.
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Poluçeno 23.02.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
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| id | umjimathkievua-article-3617 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:50Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3d/156e82fbdf78be5acdab59fb544a813d.pdf |
| spelling | umjimathkievua-article-36172020-03-18T20:00:05Z On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point О теореме Мальмквиста для решений дифференциальных уравнений в окрестности изолированной особой точки Mokhonko, A. A. Мохонько, А. А. Мохонько, А. А. The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity. Твердження теореми Мальмквіста (1913) про ріст мероморфних розв'язків диференціального рівняння $f' = \cfrac{P(z, f)}{Q(z, f)}$, де $P(z, f), Q(z, f)$ — поліноми по всіх змінних, доводиться для випадку розв'язків з ізольованою особливою точкою в нескінченності. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3617 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 505–513 Український математичний журнал; Том 57 № 4 (2005); 505–513 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3617/3965 https://umj.imath.kiev.ua/index.php/umj/article/view/3617/3966 Copyright (c) 2005 Mokhonko A. A. |
| spellingShingle | Mokhonko, A. A. Мохонько, А. А. Мохонько, А. А. On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title | On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title_alt | О теореме Мальмквиста для решений дифференциальных уравнений в окрестности изолированной особой точки |
| title_full | On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title_fullStr | On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title_full_unstemmed | On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title_short | On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point |
| title_sort | on the malmquist theorem for solutions of differential equations in the neighborhood of an isolated singular point |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3617 |
| work_keys_str_mv | AT mokhonkoaa onthemalmquisttheoremforsolutionsofdifferentialequationsintheneighborhoodofanisolatedsingularpoint AT mohonʹkoaa onthemalmquisttheoremforsolutionsofdifferentialequationsintheneighborhoodofanisolatedsingularpoint AT mohonʹkoaa onthemalmquisttheoremforsolutionsofdifferentialequationsintheneighborhoodofanisolatedsingularpoint AT mokhonkoaa oteorememalʹmkvistadlârešenijdifferencialʹnyhuravnenijvokrestnostiizolirovannojosobojtočki AT mohonʹkoaa oteorememalʹmkvistadlârešenijdifferencialʹnyhuravnenijvokrestnostiizolirovannojosobojtočki AT mohonʹkoaa oteorememalʹmkvistadlârešenijdifferencialʹnyhuravnenijvokrestnostiizolirovannojosobojtočki |