Asymptotics of the logarithmic derivative of an entire function of zero order
We find asymptotic formulas for the logarithmic derivative of a zero-order entire functionf whose zeros have an angular density with respect to the comparison function $v(r) = r^{\lambda(r)}$, where $λ(r)$ is the zero proximate order of the counting function $n(r)$ of zeros of $f$.
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| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4580 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510726539444224 |
|---|---|
| author | Zabolotskii, N. V. Заболоцький, М. В. |
| author_facet | Zabolotskii, N. V. Заболоцький, М. В. |
| author_sort | Zabolotskii, N. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We find asymptotic formulas for the logarithmic derivative of a zero-order entire functionf whose zeros have an angular density with respect to the comparison function $v(r) = r^{\lambda(r)}$, where $λ(r)$ is the zero proximate order of the counting function $n(r)$ of zeros of $f$. |
| first_indexed | 2026-03-24T03:01:35Z |
| format | Article |
| fulltext |
Y~]K 517.54
M. B. 3a6o:iom, roati (~bBia. yu-T)
A C H M I I T O T H K A 2 I O r A P H | I IOXI~IHOi
I.U2IOi | HY2II, O B O r O I I O P ~ K Y
We find the asymptotic formulas for logarithmic derivative of a zero order entire function f whose zeros
possess the angular density with respect to the comparison function v(r) = r x(O, where ~,(r) is a zero
proximate order of counting function n (r) of zeros of f.
3na~JIeHo aCHMnTOTHqlli qbopMy/m Jl/la JIorapHdi)Miqaoi noxiilaoi uiJloi qbyHgRii f HyJIbOB0*
ro nopaaKy, HyJli aK0i MalOTb KyTOny tttLqbHiCTb Bi~HOCItO qbyHKuii nopieHaaHa v(r) = rXCr), ae
L(r) - - tiyabo~a~t yTo~Hemfft nop~noz paxymnoi qbyHKLtii n (r) HyJzia f.
Hexa_tt f ~ ~ina qbyHzLIis ]IO~aTHOrO nopa~cy aiaz0M perynapHoro 3poCTaHH~ (U.
p. 3p.) S po3yMiHHi B. ~t. JIeBiHa -- IlqbJnorepa. HpHnycKaeM0, U~O n~ran 3Ha~i0M~t 3
TepMiHOJzoriem i OCH0SnaMH qbazTaMH Teopii tdJmX qbyHKui~t U. p. 3p. [1]. ~JZn
TaKHX cl)yHzt~ift f B [2, 3] 3Ha~]~eHO acrlMnTOTrlqrli qbopMyHH iX HoraprlC13MiqHrtX
noxigHnX F = f ' / f 30BHi ~eamrtx BrmaTmOBttX MmOmnH. He 3Menmyroqn 3araHb-
HOCTi, 6y~teMO aBa~aTa aa~aHi, If_tO npOMiHl, arg Z = -re e 3BHqaI~iHHM /IYI~[ HyHiB f
[1, C. 125]. Hexa~ A(V) ~ dpyHKtdJ~ KyTOBOi miHbHOCTi rIOCHi/IOBHOCTi HyHiB f,
TO6TO/~.TIg Bcix lq/, KpiM MO~YmBO He 6iHbuI aix~ 3Yti'teHHoi MHO~KI,IHH E C [-/~, 7g) ,
icHyr rparlHtOt
A(q/) = lira n ( r , - ~ , V ) -r~<~F<r~, A(-r~) = 0,
r ~ = ' V ( r ) "
ze n(r, oC, ~) ~ aHCHO HyHiB f B CeKTopi {Z: Iz l -< r, a -< argz < [~}, V(r) = r p(r),
p (r) ~ yToaHeHatt nopa~ao~ f. 3ayBa)znMo, mo Heo6xi/IHOm Ta aocTaTn~Om
yMOBOIO 14. p. 3p. ~yHKraii f tteuiHoro nopJzaKy r icHyBanaz KyTOBOi miHbHOCTi ii
Hyzi~ Bi/~HOCHO cl0yHKl-tii IIOpiBH~IHHZ V ( r ) .
Y Brlria/~Ky, KOHrl nopg/~OK qbyHKlfii f /~opiBI-llOr HyHIo, yToqaeHrltt llOp~-
aOK p (r) ~yHKLdi f B~Ke He 6y/~e yTOqHeHHM FIOpSIJ~KOM pax3qo~oi qbyaKI~ii n ( r ) =
= n ( r , -rr /~) ii ltyaiB. 5[K noKa3arlo s [4], TOni n ( r ) = o ( V ( r ) ) , r --> oo. OT ~e ,
ay~xi I~iHoi dpy~Kuii ttyH~oBoro r m p ~ y 3a~7,r 6y~XyTb MaTH KyTOBy II~i.rlbHiCTb
A(V) = 0 Bi~aOCHO qbynKraii nopiBnmma V(r). ToMy y BIma~Ky HyH~,OBOrO nopza-
Ky pozrH.a~aeMo cl)yH~ttim nopiBrumHz v (r) = r x(r), v (0) = 0, ~e
1) dpyHr,~ia 3, ( x ) ~ HeBi~'r Henepep~Ho ~Hqbepemtii~oBHa Ha [0, +o.) ;
2) k ( r ) --> 0 npa r --~ +o,;
3) e(r )=k(r)+rk ' ( r ) lnr- ->O npa r-->+~o;
4) v ( r ) $ + * * rIpa r -*+**;
5) 0< lira n(r)/v(r)<+o*.
r--~ +m
(~yHKI.LilO ~,(r) Ha3HBalOTb HyYlI=OBHM yTOqI-IeHHM nop~l~ZOld qbyrlxtdi n ( r ) .
Kpit4 qbyHKaii nopisHmm~ v(r) 6yAeMo po3r~igAaTrl TaKOat qbyHKttiIo ~(r) = r x(r),
u(0 ) = 0, Ae ~(r) ~ CHYlI=HHtt HyYmosait yTOqHeHrllt nopaaoK n ( r ) , TO6TO
[l , c. 56 - 60]
�9 M.B. 3ABOJIOIIbKH~, 1999
32 ISSN 004 ! -6053. Ygp, ~tam. ~.'vpn., 1999, m, 51, N e 1
ACHMHTOTHKA JIOFAPHOMIqHOI I-IOXI~HOI l.[IflOl OYHKIIII ... 33
~(r) = O(ln r)/ln r. (1)
TyT O ( x ) - BrHyTa qbynKtli~I, IIIO 3a~oao~Ibn~le HaCTyrlHi yMOBI4:
a) O ( x ) $ + ~ npn x - - > + ~ ;
6) O(x)/x-->O npu x-->+~;
B) O"(x)/O'(x) ~ 0 rtpu x--~ +~;
r) 0< llm n(r ) /~ ( r )<+~.
r ---> -[- oo
Bi;~oMo [1, c. 60], mo ~ a ~oai~bnOi neo6Me~enoi 3Bepxy ~O~aTHoi na [ 0, + ~ )
qbyHKIAii r uy~oBoro n0pailI<y icnyr ii cnamHn~ yToqaenI4~ nop~oK ~(r).
Y pO60Ti 6y;~yTb 3uafl~eni acm4nTOTh-qni qbopMy~n ~ a
F ( z ) = f ' ( z )
f (z ) "
~e f ~ rti~a qbyHKuia ny~boBoro n0pz~Ky, ny~i aKoi ~aZOTb XyTOBy miasnicT~
A(Xl/) Bi/~HOCHO qbyHKUii rIOpiBHJIHHJ~ V (r) a6o 5(r).
3ayBaYX;IMO, ~Xmo ]~YI~I 12iJmx qbyrIKtfill tt. p. ap. acnMnTOTrIKy In If(z) [ MO~KHa
6yno BKa3aTH 3OBHi /~e$1KO1 IvIHO~KHHH Kpyria HyYlbOBOI niHit~H0i miYr~nocTi, TO/~nJ~
F(Z) tle, Baarani Ka~KyqH, HeMo~nnBO HaBiTI, a naflnpocTituoMy Bnna/IKy LfiJ~oi
qbyHKnii Hettinoro nopa~axy 3 HynaMn Ha o]~noMy npoMeni [2, c. 63].
Hexatt aa/xaHa CnCTeMa K KpyriB {Z: Iz--aj l< 9 }, j = I, 2 . . . . . aj ---> 00, j --->
--~ +oo i I~ - - ~eaKe ~ncno, I < ~t _< 2. HKmO
lim r -~ E ~ = 0 ,
r--~+~ [ajl<r
TO Ka.>KyTb, 1-120 CHCTeMa KpyriB K ~ae nyYraoBy IS-mi~buiCT~. 3ayBaaca~o, mo BeCb
npoMinb arg Z = a MO~KHa noKprlTrl CHCTeMO~ K Kpyria HyYlbOBOi p.-uliYIbHOCTi,
1 <IX_<2.
CnpaBeI~nHai HacTynHi TeopcMH.
TeopeMa 1. Hexaa f m tr q~ynmli,~ ny.at,oeoeo nop,~Oxy ~ eiO'e~mu~m ny.a.a~tu,
a > O, ~(r) - - qb)'mc~i~ onyrna oiOnocno aoeapuqb~o', ~(r) = ~(r) + r ~'(r) In r.
.Clrugo
n(r) = A~(r)+o(~(r)~(r)), r ~ + ~ , (2)
mo Onz ~cix z = re iO, - n < O < ~ ,
F(z) A~(r) + iA~;(r)~(r) 0 = ~ + o r--->+**. (3)
z Z
Hpu ~bO~t)' cnia~iOnomenn~ (3) ouronyemw~ piono~dpno ei~nocno 0 a Ooai~no~ty
h 3 , m i - ~ + 5 < O < g - $ , 0 < 5 < 1 .
TeopeMa 2. Hexaa f ~ ~ina qbynrt4i~ nyn~oaom nop,~')', nyai mwi" ~tatomb
xymoay u4in~nicm~ ~iOnocno ~yn~ii" nopiettannn v ( r ). ToOi icnye cucme~ta xpyeia
E nyn~oao~ ~t-u4inbnocmi ma~, u4o npu Z e r au~nyemw~
F(re iO) = A v ( r ) / z + o (v ( r ) / r ) , (4)
piono,~tipno ~iOnocno 0 npu r --> + ~.
ISSN 0041-6053. Yrp. atom. ~.'vpn., 1999, m. 51, N e 1
34 M.B. 3ABOJIOU.bKHI~
3ayaaxaMo, mo B [5] oAepacaao acaMrtTOTtlqHi qbopMy~m ~ana In f ( z ) ui~oi
qbyur, aii f ny~a, oaoro nopJcazy, ny~i azoi si~'r ra 3a~oao~,nmOT~ yMoBy (2).
~rt,q ROBe/~eHI-I~I l!yIX TCOpCM BI4KOpPIcTaelv[O nacTynni aezn.
J'Ievta 1. Hexad X(r) ~ cunbnua ny,,~o6ua ymo~nenua nopaOo~. ToSi
~(r) = O ' ( l n r ) --->0, r - - ,+o . , (5)
d~(r) 3(r)~(r), d2u(r) = o(D(r)~(r)), r --> +00.
d ln---'-~ = d( ln r) 2
t~Iooe~enna. 3 (1) oAepxyetao, mo ~(r) = O ' ( l n r ) . OCKi:mKH O ' ( X ) <
< ( O ( 2 x ) - O (x)) /x, TO 3as]~JtKrl yrdosi 6) cHJzbnor0 nyJIbosoro yToqnenoro no-
pJ~aKy ~(r) 0TpttMyCMO (5).
]~ani ~'(r) = (?;(r)l r)~(r), TO~ty
dT~(r) = ~(r)~(r).
din r
BpaxoaymqrI (5) i yMosy B) CtlJIbHOFO ayJmosoro yTOqrleHOI'O nop~Ky, MaeMo
e - ~ O: = ~(r)~Cr) ~(r) + =
- I =
JIe~j 1 ~Iose~leno.
JIe~a 2. Hexaa ~ ( r ) ~ s p o c m m o ~ a ~a [0, +~,) dp3'n~gia, ~ ( r ) l r ~ cnaOae
Oo ~yn~ npu r --*+~, 0 < y < 1. ToOl 8 ,~ Ooeim, noi'imnezpoenoi'dpyn~ir [3(r),
~(r)--r npu r-->+** iO~a z = r e i8, - g < 0 < r c ,
[ ( z+ t )~_ ) = o , , - o + o ~ ,
npu~o~ty tla otlinra piono~dpna oiOnocno 0 e Ooeim, no~o' ~ymi [-~t + 5 < O < rc -
-8], 0<8<~
J-le~a 3. Hexa~ ~(r) ~ qbyn~ia, onyrAa ~iD~ocno ,~ozapu~so', 0 < ~ < I,
(l-8) r r
~zk(r,e ) = f [;(t)~(t) tkdt, ~zk(r) = ~[;(t)~(t)tkdt,
0 0
+o0 +00
bl(r, ~) = ~D(t)~(t)t-k-2dt, gk(r) = ~D(t)~(t)t-k-2dt.
( l + e ) r r
ToOi Oaa z= re i~ -Ic < 0 < g,
+~ +=o
K" t t,k_-i0(k+l)~lk(r) lim ~ (--1)ke -i0(k+t) ~k(r' g)
"- 'ok =o ~ = k--oLt-" ~ ~ ,
+" /;k(r, g) + " , ,,k _iO(k+l)/;k(r)
lim ~ (-1) ke/0(k+l) = ~ t - , , ~ r_---f~. r-k+l
(6)
(7)
ISSN 0041-6053. Y~p. ~am. ~.'ypn,, 1999 , m, 5 l , N e I
ACHMrlTOTHKA $1OFAPHOMIHHOI FIOXH2HO112I./IOI OYHKIIII ... 35
dIena 4. Hexaa rl ( t ) ~ inmezpoona cib)'nrt(i.~ na [0, +.o). r I ( t ) --40 npu t ---)
+0% ~(t) mara , ~ o need 3,
ToOi
r + . o
Ak(r) = ~ f ~(t)~(t)l~(t)ttdt, Bk(r) = l--L- f ~)(t)~(t)rl(t)t-k-2dt.
k + l k + 1 0 r
+** ( l)k e_iO(k+l ) At(r) E; = E - rk+l = o(~Cr)~(r)), r ---) +0%
k=0
�9 .I- e o
r_k+ 1 = o(~(r)~(r)), r---> +~.
k=O
(8)
(9)
~OBe/IeHHJ~ JleM 2 - 4 He HaBO,/~HMO, OCKi.rlhKH BOHO aHaJIoriqHe /~OBe/IeHHIO
i 3 :aeM - at31.
d-IeMa 5. Hexa~ ouKony~ombc.~ y~toeu .r 3 i
+oa
Y'I = ~ (--l)k e-iO(k+l)~tk(r) r-k-l '
k=O
+ ~
Y'2 = ~ (--l)k eiO(k+l)~)k(r)rk+l"
k=0
~lk ( r ) =
AHa~oriqHo
Toai 0.r z = re iO, - i t < 0 < g,
- Y I + Y2 = iO~(r)~(r)+o(~(r)~(r)), r-++oo. (10)
, l lo6eaenu~ OcKi~bKrl 3a ~eMO~O 1
(V(t)~(t))' 1 d2v(t) rl(t)v(t)~(t)
= t d ( ln t) 2 - t '
/le rl (t) ---> 0 npri t --~ +0% TO npoinrerpyBaatmt qacTrmaMri, o~Iep~yeMo
r
?)(t)~(t) tk+l I r . 1 . [ (B(t)~(t))'tk+t dt = ~;(r)~(r)r k+l ~k(r).
k + l 0 g + l ~ k + l
v(t) 8(t)~(O , ~)(r)~(r)r-k-l + f - - ~ d , =
[Tk(r) = k + l
r
= ?;(r)~(r)r-k-I - B~(r).
k + l
~asfi , SpaX0BylOqH (8) Ta (9), orpnMyeMo
+'* ( 1,keiO(k+l ) 1 ~;(r)~(r)
+ * a
--~'1 + ~2 = -- ~ (--1) ke-iO(t+l)~;(r)~(r) +
k + l k + l
k=O k=O
+ ~ + Y2 = ~(r)~:(r)(- ln ( l + e - i ~ ( l+eiO)) + o(~(r)~(r)) =
= ~(r)~(r)(ln e iO +o(1)) = iO ~(r)~(r)(1 +o(1)), r---> +0%
I/J[O ,/IOBO.//,I, ITb ~eMy 5.
+
ISSN 0041.6053. Yrp. ~gtm. ~.vpu., 1999, m. 51. bl ~ 1
36 M.B. 3ABOJIOIAbKHI~I
d'IeMa 6, Hexaa v ( r ) = r ~(r). 8(r) = ~,(r) + r~, ' (r) lnr,
r +**
ak(r ) = Iv(t)~(t)tlCdt, bk(r) = Iv( t )~( t ) t -k-2dt .
o r
ToOi Oa~ O, - ~ < e < re,
F'3 = Z (-1)ke-ie(k+l)r-k-lak(r) = o ( v ( r ) ) , r---> + ~ ,
k=O
+oo
Z4 = Z (-1)keie(k+l)rk+lbk(r) = o ( v ( r ) ) , r---> +*%
k=0
npuno~ty cni6eianoutenn~ (11)ma (12) auKonytombc~ piono;~dpno eiOnocno
-re+ ~< O<rc-5, 0<5<1.
]Ioeeaenn~.. Hexalt
[E31 = ,k=O+~(--l'ke-iO(k+l)r-k-l(r!+rd/2) t_krf)v(t'8(t' Idt <
)r,:
_< v(rr) 2 -k IE(t)ldt +
\k=o Y o
OCKiJIbKI4
+
+ - : ~ h I J'~ I = ' + ''='
k = Ok r12 J
r12
1 ~ le(t)ldt = o(1),
r
0
r ~ +oo,
-~-oo
~ 2 -k = 2 ,
k=O
TO J L = o ( v ( r ) ) , r--->+~o.
I'[OKJIa~eMo
g(r) = sup{l~(t)[: 2<t<r},
r
I (1 )ke- iO(k+l )
uk(r ) = "Tg-f I v(t)r r = -
r r12
MacHo
r r~12 kr J
TO6TO (uk(r))~ 0 --cna~aa noani~onnicTb, a
<0,
k(O) = ] l + e :i0 [ < sin(-8/2)
(11)
(12)
o,
ISSN O041-6053. Yrp. ~u~n. acy. pn.,1999 , m. 51. Iv~ 1
ACHMI'ITOTHKA ~OFAPHOMIqHOI HOXI~J-IOI IB.rIOI OYI-IKI.UI ... 37
]]erIJl Bcix n > 0 i ~I~ BCiX O ~ [--g + ~, g + ~] . TaKHM tlHHOM, 3a oarIaICom ~Iipix~e
pe, z~
14 = Z ut(r)fDk(O)
k=0
piBno~ipno 36ixc~nlt ni/~ocno 0 e [-re + ~, ~x - ~] . ~a~i, za HepiBHiCTm A6eaa,
IS~(r ,O)l -- u~(r)m~(O) _< o~(0) ( l~ ( r )+21un(r ) l ) <_
k--O
g:(r) v(r)
-< sin(~/2) ( l + n 2 + i ) ,
aairtzrioiaepxyeMo, mo 12= lira S n ( r , O ) = o ( v ( r ) ) , r--->+**.
n - - . ~
~0Be/~eMo piBHiCTb (12). HoK.rta~eMo e*(r) = sup {[e(t)[: r < t < 2r},
2r
u*~(r) = rk+l f v ( t ) e ( t ) t - t -2d t , c0~(0) = (-1)ke i0(t+l).
r
Toni
( ; 11 [Z4[ = co~(O) u~(r)+r k+l + v(t)e(t)t -k-2 dt <_
�9 2r
I I; 01 < tO*k(O)u~(r ) + 2 -k r v(t)F.(t)t -2 dt = I J31 + J4 o
2r
OCKiSmKrI irrrerpa~
+ * o
l ( r ) = f o(t)~.(t)t -2 dt
2r
36ia<Hrli~, TO 3a ~aonoraorom npaBrma J-Ionira.az nesaxzo noza3aTi~, mo r l ( r ) =
= o ( v ( r ) ) , r.-->+o,. 3Bi/Icrl J 4 = o ( v ( r ) ) , r.-.>+oo.
P ~ J3 oaiaIor aHa~ori'~no p~ay J2- IIOC0Ii]IoBHic'rb (uk(r))k+__O cnatma,
qaCTKOBi cyMa pa/Iy EC0~(O) 06Me~KeHi B cyKynHocTi a a a 0 e [ - ~ + ~, x + ~] ,
k=0
a OTT~e, pz~ J4 piano~ipHo a 6 i ~ n i l ni~nocHo 0. ~asti
[S.Cr, O) = u (Ocoi(o) <
2 )
+ <_
v(2r) e*(r) )
'~ ~ +** 2 sin (fi / 2) '
a s i ~ o~epmye~o, mo J4(r) = o ( v ( r ) ) , r--> +**. J'IeMy 6 ~ose~eao.
~oeeaenna meope~u 1. Hexatt f m uiaa qbyuKttia HyJlsoBoro n o p a ~ y ,
(-an) --nocJfi~OBHiCTb ii HyJfin, an> 0, n r H , mo 3a~oBosmHar yMoey (2). Toni
py,~ Z+~o (Xlan) 36imm~i~J~ z = r e '0, -~<0<~, .ae~o
ISSN 0041-0053. Yxp. 7uam. a,.vp,., 1999, m. 51,1~ 1
38 M.B. 3ABOJ'IOI2bKHfl
F(z) = ~ 1 dn(t) n(t) ** n( t )dt
k=lZ+a-"~ = ~ Z'+';" = t"+Z + (t+O 2 -
+" " T :. = f n ( t ) -A '~ (t)dt + = II +
o (z+t) ~o
lloKna~eMo a ( t ) = [;(t)~(t), [~ (t) = (n( t ) - AT;(t))la(t). To•i, aaB~aJ~Kn onyKnocTi
di3yHKllji V(t) Bi/IHOCHO norapHqbMy, netai 1 Ta cniBBi~aaomeHrzIO (2), C~yHKI.Ii~I (~ (t)
6yae 3pocTazOqOZO, a [] ( t ) ~ 0 Izprz t ~ +~0. 3a neMom 2 oaepmye~o II =
= oCfi(r)~(r)/r), r---> +~.
I h z i
'2 = A ( l + t ) - + ! t 2 ( l + Z ) 2
,s :::" "r " ::I)' = A 1 D( t ) (-1)k(k+l) + D ( t ) t - 2 ~ ( - l ) k ( k + l ) d t =
r k=0
s ": 1
(l-~)r +o.
= _ (-l)k(k + 1) ~(t)t-k-2dt . (13) A lira (--1)k(kzk+~ +1) ~(t)tkdt + ~ , , Z -k-~
Z ~--~0 0 k=O (l+~)r )
OCKiJISKH
(l-~)r I
I I [ ~(O~(O:d, = I ;(t)tkdt = ~(t)tk+l (I-r (1-~)r
0 k + l ~0 k + l 0
3((I - ~) r)((I - E) r) k+1 I
- ~k(r, s)
k+l k+1
+" ~((: + e) ~).((1 + e) r)-k+.[ + :+ _
I v(t)t-k-2dt = k+l k i bk(r'e)'
([+e)r
TO 3 (13), ~,aB~t~H (6) Ta (7), oTprZMycMo
12 = ~ lim (k~__~O (-l)k(l- e)k+l +**(--l)keiO(k+l)')-
z e-~o e ie(k+') '' + k=02 ~--e'~'~T jr(r) +
A( ~--~ (-l)k ~ . , +** k )
+ -- |- L -Sk--~-aktr) + ~o~bk(r) =
Z ~ k=O z =
= AD(r) lira l - e 1 + l + e 1 ) A(_Zt+Z2 )
z ' ~ - ~ ~ -/~ e--'=Tffl+(l+e)e io + z =
= a~(r) + ~(-z, + z~).
z z
3Bi~Crl, Bpaxoayzoqrl (10), MacMo (3), tUO aoaotma'~ Te0peMy 1.
(14)
ISSN 0041-6053. Yrp. .uam. ~.'yp,., 1999, m. 51,1~ 1
ACHMHTOTHKA JIOFAPHOMIttHOI HOXIL1HOI HIJ]OI OYHKL[II ... 39
]~7[oeeaeuu~ meope~u 2. Hexa~ f - - uiJla ~yHKtfia nop~tKy HyJm, HyJfi aKOi
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ISSN 0041-6053. Yrp. ~u~m. ~.Tpn,, 1999, m. 51, IW 1
40 M.B. 3ABOJ'IOI.I, bKHI~
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r e M i n p o r d e a i a s n a c J z i ~ z y ~ie Mo~ceMo, r o ~ y tUO a ~ a n o r a n p o z c m d a t f i i a n o i s~erdn 1 3
[2] t~ar H a 6 J m ) z e a a J z F ( z ) a " roqa i c r IO o(D(r) /r) , a a e a " r o ~ - f i c r ~ o(~(r )~(r ) l r ) ,
r -.--y -t- o~.
{z: arg z = ~p}, 1"o, a z i
1. J'le6un B. ~. Pacnpcae~qemte zopnelt UeJIblX qbytiKl~l,l~t. -- M.: FHTTJ'I, 1956. - 632 c.
2. Fon~96ep~ A. A., Kopenro6 H. E. ACrlMnTOTPIKa Jlorapr~qbMxqeczo~l npoHaBo~tuoll ue~oia qbyHz-
urm ano~me pery.~apHoro poera//Cr16. MaT. a<ypn. - 1980. - 21, N ~ 3. - C. 63 - 79.
3. Foa~6epe A. A., Kopenroo H. E. 0 6 aeHMn'roTnKe JloraprtqbMnqeeKog npona~o~Uoia tte.qo~l
qbynztm~t BnoJme pcry~apuoro p o e r a / / Y z p . Max. a<ypu. - 1978. - 30. N*- 1. - C. 25 - 32.
4. Fonbb6ep~ A. A., 3a6ono~ru~ H. B. Hn~texc KO~fUenTpatmx cy6rapMormqeczott qbyuzttt~a nyJle-
aoro nopa/!za H MaT. ~a~e'm~. - 1983. - 34, N TM 2. - C. 227 - 236.
5. 3a6oao~cu~ M.B. TeopeMn TMny Banipona xa Ba.nipoHa - TiTqMapma/1i!~ ttiJmx qbynztti~t
ny~bosoro nopz~tzy H YKp. MaT. )zypn. -- 1996. -- 48, l ~ 3. -- C. 315 - 325.
O~ep~xano 30.01.97
ISSN 0041-6053. Yrp. ~tam. acypH., 1999. m. 51, N e I
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| id | umjimathkievua-article-4580 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:35Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/0f4de46a26f64b11607828f0c405fc90.pdf |
| spelling | umjimathkievua-article-45802020-03-18T21:09:14Z Asymptotics of the logarithmic derivative of an entire function of zero order Асимптотика логарифмічної похідної цілої функції нульового порядку Zabolotskii, N. V. Заболоцький, М. В. We find asymptotic formulas for the logarithmic derivative of a zero-order entire functionf whose zeros have an angular density with respect to the comparison function $v(r) = r^{\lambda(r)}$, where $λ(r)$ is the zero proximate order of the counting function $n(r)$ of zeros of $f$. Знайдено асимптотичні формули для логарифмічної похідної цілої функції f нульового порядку, нулі якої мають кутову щільність відносно функції порівняння $v(r) = r^{\lambda(r)}$, де $\lambda(r)$—нульовий уточнений порядок рахуючої функції $n(r)$ нулів $f$. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4580 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 32–40 Український математичний журнал; Том 51 № 1 (1999); 32–40 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4580/5864 https://umj.imath.kiev.ua/index.php/umj/article/view/4580/5865 Copyright (c) 1999 Zabolotskii N. V. |
| spellingShingle | Zabolotskii, N. V. Заболоцький, М. В. Asymptotics of the logarithmic derivative of an entire function of zero order |
| title | Asymptotics of the logarithmic derivative of an entire function of zero order |
| title_alt | Асимптотика логарифмічної похідної цілої функції нульового порядку |
| title_full | Asymptotics of the logarithmic derivative of an entire function of zero order |
| title_fullStr | Asymptotics of the logarithmic derivative of an entire function of zero order |
| title_full_unstemmed | Asymptotics of the logarithmic derivative of an entire function of zero order |
| title_short | Asymptotics of the logarithmic derivative of an entire function of zero order |
| title_sort | asymptotics of the logarithmic derivative of an entire function of zero order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4580 |
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