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Асимптотичні властивості аналітичних та субгармонійних функцій нешвидкого зростання: Автореф. дис… д-ра фіз.-мат. наук / М.В. Заболоцький, Львів. де

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Euea?anueeee aea?aeaaiee oi?aa?neoao ?iai? ?aaia O?aiea

Caaieioeueeee Ieeiea Aaneeueiae/

OAeE 517.535

ANEIIOIOE*I? AEANOEAINO? AIAE?OE*IEO OA

NOAAA?III?EIEO OOIEOe?E IIA?EUeIIAI C?INOAIIss

01.01.01 – iaoaiaoe/iee aiae?c

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Euea?a – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia o Euea?anueeiio aea?aeaaiiio oi?aa?neoao? ?iai? ?aaia
O?aiea ia eaoaae?? oai??? ooieoe?e ? oai??? eiia??iinoae.

Io?oe?ei? iiiiaioe:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Oai?acia I?iia?c Iae?eiae/,

?inoeooo iaoaiaoeee IAI Oe?a?ie,

caa?aeoaa/ a?aeae?eo eiiieaeniiai aiae?co oa oai??? iioaioe?aeo;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

N?aee?oeeee Aiaoie?e Ia/eneaaiae/,

Iineianueeee aea?aeaaiee oi?aa?neoao ?i. I.A.Eiiiiiniaa,

i?ioani? eaoaae?e iaoaiaoe/iiai aiae?co;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, aeioeaio

Aeiieoeueeee Aiaaeai Aaneeueiae/,

Ae?iaiaeoeueeee aea?aeaaiee iaaeaaia?/iee oi?aa?neoao,

i?ioani? eaoaae?e iaoaiaoe/iiai aiae?co.

I?ia?aeia onoaiiaa Oa?e?anueeee aea?aeaaiee oi?aa?neoao, eaoaae?a oai???

ooieoe?e ? ooieoe?iiaeueiiai aiae?co.

Caoeno a?aeaoaeaoueny 20 o?aaiy 1999 ?ieo i 15.20 aiae. ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 35.051.07 o Euea?anueeiio aea?aeaaiiio
oi?aa?neoao? ?iai? ?aaia O?aiea ca aae?anith: 290602, i. Euea?a, aoe.
Oi?aa?neoaonueea, 1, Euea?anueeee aea?aeaaiee oi?aa?neoao ?iai? ?aaia
O?aiea, iaoai?ei-iaoaiaoe/iee oaeoeueoao, aoae. 377.

C aeena?oaoe??th iiaeia iciaeiieoenue o Iaoeia?e a?ae?ioaoe?
Euea?anueeiai aea?aeaaiiai oi?aa?neoaoo ?iai? ?aaia O?aiea (aoe.
Ae?aaiiaiiaa, 5).

Aaoi?aoa?ao ?ic?neaii ” 17 ” ea?oiy 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee __________ ss.A. Ieeeothe

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Eeane/i? oai?aie TH.Nioioeueeiai-O.Kaci?ao?
(1868?.), E.Aae??oo?ana (1876?.) ? A.I?ea?a (1879?.) caii/aoeoaaee
oai??th ?iciiae?eo cia/aiue ia?iii?oieo ooieoe?e. A 90-eo ?ieao OIO
noie?ooy ? ia ii/aoeo OO noie?ooy aeacai? oai?aie io?eiaee na?e
iiaeaeueoee ?icaeoie a aeine?aeaeaiiyo ?iciiae?eo ioe?a oe?eeo ooieoe?e,
i?iaaaeaieo a iniiaiiio o?aioeocnueeith oeieith (AE.Aaeaia?, A.Ai?aeue,
A.E?iaeaeueio, AE.Aae??ii oa ?io.). Aiae?oe/iee aia?ao aeey ia?iii?oieo
ooieoe?e aoa iiaoaeiaaiee a 20-eo ?ieao iaoiai noie?ooy o?inueeei
iaoaiaoeeii ?.Iaaaie?iiith, i?ney i?aoeue yeiai oai??y ?iciiae?eo
cia/aiue iaaoea iaaiiai caaa?oaiiai aeaeyaeo. C oeueiai /ano oey oai??y
caeia? iaeia c oeaio?aeueieo i?noeue a caaaeuei?e oai??? ooieoe??
eiiieaenii? ci?iii?.

Ia caaaeath/e ia iaaio caaa?oai?noue oai??? ?iciiae?eo cia/aiue,
aea/aiiy iaa?oue eeane/ieo ?? caaea/ ua ia aeiaaaeaii aei e?ioey.
Caeeoa?oueny a?aee?eoei ?yae ieoaiue ia?oi/a?aiai? aaaeeeaino?, a aei
ieo aeieo/athoueny iia?, ui aeieeathoue a i?ioean? aeine?aeaeaiue.

Iniaeeao oaaao a inoaii? ?iee i?eae?eythoue aea/aiith niaoe?aeueieo
eean?a oe?eeo oa ia?iii?oieo ooieoe?e, ye? aecia/athoueny /e oi
iaiaaeaiiyie ia c?inoaiiy oeeo ooieoe?e, /e ?aaoey?i?noth iiaiaeaeaiiy
iaaaie?ii?anueei? oa?aeoa?enoeee, /e ?icoaooaaiiyi a-oi/ie oaeeo
ooieoe?e.

A oai??? oe?eeo oa ia?iii?oieo ooieoe?e /anoi cono??/athoueny neooaoe??,
eiee iaaaie?ii?anueea oa?aeoa?enoeea ? iia?eueii c?inoath/ith.
Aneiioioe/i? aeanoeaino? oaeeo ooieoe?e iiae?ai? aei iiaiaeaeaiiy
iiiai/eai?a, ? oe? ooieoe?? ? aeno?aiaeueieie a aaaaoueio caaea/ao
oai??? ?iciiae?eo cia/aiue. Oiio i?e?iaeiei ? ieoaiiy, eiee iniiai?
oa?aeoa?enoeee oe?ei? ooieoe?? (eiaa?eoi iaeneioio iiaeoey,
iaaaie?ii?anueea oa?aeoa?enoeea, eiaa?eoi iaeneiaeueiiai /eaia oa
iaaaie?ii?anueea e?/eeueia ooieoe?y ioe?a) ? iia?eueii c?inoath/eie. Oea
ae noino?oueny ? ooieoe?e aiae?oe/ieo a e?oc?.

Iaei??th c iniiaieo caaea/ oai??? oe?eeo ooieoe?e ? aea/aiiy ca’yce?a
i?ae ?aaoey?i?noth c?inoaiiy ooieoe?? oa ?aaoey?i?noth iiaiaeaeaiiy ??
e?/eeueii? ooieoe?? ioe?a. A 1914 ?ioe? AE.Aae??ii anoaiiaea oaeee
ca’ycie aeey oe?eeo ooieoe?e aeiaeaoiiai ii?yaeeo c ioeyie ia i?iiai?. A
1936-1939 ??. A.Eaa?iei oa A.Ioethaa?ii aoea noai?aia oai??y oe?eeo
ooieoe?e oe?eeii ?aaoey?iiai c?inoaiiy (oe.?.c?.) ne?i/aiiiai
iaioeueiaiai ii?yaeeo. Aeooaeueiith ? iiaoaeoaa aiaeia?/ii? oai??? aeey
oe?eeo ooieoe?e ioeueiaiai ii?yaeeo.

Aea/aiiy aeanoeainoae ia?iii?oieo ooieoe?e a oa?i?iao c?inoaiiy
noa?e/ii? iio?aeii? ii/aeinue a 20-eo ?ieao iaoiai noie?ooy a i?aoeyo
I.AEthe?a, A.Ino?ianueeiai, O.Ia?o?, E.?in?aee ? io?eiaei iiaeaeueoee
?icaeoie a i?aoeyo I.Eaooi ? E.A??oaiaia (1951-1959??.) oa A.?.Aaa?eeiaa
(1965-1968??.), yeee cai?iiiioaaa eeaneo?eaoe?th ia?iii?oieo ooieoe?e ca
c?inoaiiyi noa?e/ii? iio?aeii?. Ieoaiiy i?i ca’ycie noa?e/ii? iio?aeii?
c iaaaie?ii?anueeeie aaee/eiaie aea/aeinue aaaaoueia
iaoaiaoeeaie(A.A.Aa?naayi, A.A.Aieueaeaa?a, Aeae.Aiaea?nii, E.A?ieea?,
O.Aec?iii, Aeae.Eeoi?, N.Oiii?ea oa ?io?). Oiio anoaiiaeaiiy ca’yce?a
i?ae noa?e/iith iio?aeiith, iaaaie?ii?anueeith oa?aeoa?enoeeith,
aeiyoeiaeie aeaoaeoieie cia/aiiyie, i?ea??anueeeie iiiaeeiaie
a?aeiinyoueny aei iaeano?, ui aeoeaii ?icaeaa?oueny ? ia? aaaeeea?
canoinoaaiiy.

Iiiaae i?anoie?ooy oaaao iaoaiaoee?a i?eaa?oa? caaea/a ii??aiyiiy
c?inoaiiy eiaa?eoia iaeneioio iiaeoey, iaaaie?ii?anueei? oa?aeoa?enoeee
oa aaee/ei N(r,0,Y)=N(r,0)+N(r,Y) ? N0(r)=max{N(r,0), N(r,Y)}
ia?iii?oieo oa oe?eeo ooieoe?e (AE.Aae??ii, ?.Iaee?, I.A.Aiai?ia,
A.I.Iao?aiei oa ?i.). Aeio?eioth/enue no/anii? oaiaeaioe?? ?icaeyaeaoe
oe? ieoaiiy c iaecaaaeuei?oi? oi/ee ci?o – oai??? noaaa?iii?eieo oa
d-noaaa?iii?eieo ooieoe?e, ie ?ica’yco?ii caaea/o i?i ciaoiaeaeaiiy
oi/ieo ioe?iie caa?oo oa cieco aeey aaee/ei oeio oa ieaeiueiai oeio aeey
oeeo oe?oeo eean?a ooieoe?e ioeueiaiai ?iaeo, ia?aoiaey/e, aea oaea
ocaaaeueiaiiy i?e?iaeia, ? o m-aei??iee i?ino??.

Noaaa?iii?eia ooieoe?y iiaea a aeayeeo oi/eao i?eeiaoe cia/aiiy -Y.
Iiiaeeia oaeeo oi/ie ia? ioeueiao ?ii?noue. Iaei??th c aaaeeeaeo caaea/
? ciaoiaeaeaiiy aeey noaaa?iii?eieo ooieoe?e v(z) oi/i?oeo ioe?iie
cieco, i?ae ia??ai?noue v(z)>-Y, ciai? aeiyoeiaeo iiiaeei. Oae? ioe?iee
cono??/athoueny a i?aoeyo E.Aeueoi?na ? I.Oaeina (1949 ?.), O.Oaeiaia
(1956 ?.), I.A.Aiai?iaa (1968 ?.) oa aaaaoueio ?ioeo. Aea aiie ia ?
oi/ieie aeey noaaa?iii?eieo ooieoe?e ioeueiaiai ii?yaeeo. Oiio
aeooaeueiei ? io?eiaiiy iiaeo aneiioioe/ieo oi?ioe aeey d-noaaa?iii?eieo
a Rm, m? 2, ooieoe?e ioeueiaiai ii?yaeeo aeey oiai, uia ae?noaoe oi/i?
ioe?iee cieco oeeo ooieoe?e.

Caycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Cia/ia /anoeia
?acoeueoao?a, aeeeaaeaieo o ?iaio?, io?eiaia a i?ioean? aeeiiaiiy oaie
AeEIO Io-139Ae “Aneiioioe/i? aeanoeaino? aiae?oe/ieo oa noaaa?iii?eieo
ooieoe?e i?e iaaeeaeaii? ci?iii? aei e?ath iaeano?”, aea?aeathaeaeaoieo
oai Io-202A “Oe?e? ooieoe??, ?yaee Ae???oea oa ?o canoinoaaiiy” oa
Io-380A “Aiae?oe/i? ooieoe?? oa ?yaee Ae???oea”.

Iaoa ? caaea/? aeine?aeaeaiiy.

1) Aeacaoe iaiao?aei? oa aeinoaoi? oiiae ia oaeei??anuee? eiao?oe??ioe
oe?ei? ooieoe?? aeey oiai, uia eiaa?eoi iaeneioio iiaeoey, eiaa?eoi
iaeneiaeueiiai /eaia oa iaaaie?ii?anueea oa?aeoa?enoeea aoee iia?eueii
c?inoath/eie ooieoe?yie. Oaeo ae caaea/o ?ica’ycaoe ? aeey aiae?oe/ieo a
iaeeie/iiio e?oc? ooieoe?e.

2) Aeiaanoe oaoaa?ia? oai?aie aeey oe?eeo ooieoe?e ioeueiaiai ii?yaeeo
oa aeey aeiaooe?a Aeyoea, e?/eeueia ooieoe?y ioe?a yeeo ? iia?eueii
c?inoath/ith.

3) Aaanoe iiaa iiiyooy ?aaoey?iino? c?inoaiiy oa “?iaeeeaoi?a” aeey
oe?eeo ooieoe?e ioeueiaiai ii?yaeeo oa ciaeoe e?eoa??? oe???
?aaoey?iino? a oa?i?iao ?iciiae?eo ioe?a, iienaoe eeane oe?eeo ooieoe?e
?c caaeaiei ?iaeeeaoi?ii.

4) Ciaeoe caaaeuei? oiiae ia iiiaeeio EI C aeey oiai, uia aiia aoea
i?ea??anueeith aeey eean?a ia?iii?oieo ooieoe?e.

5) Aeacaoe eeane ia?iii?oieo ooieoe?e, ye? aecia/athoueny iaiaaeaiiyie
ia c?inoaiiy noa?e/ii? iio?aeii? oa iaaaie?ii?anueei? oa?aeoa?enoeee ?
ye? ia iathoue iaaaie?ii?anueeeo aeiyoeiaeo cia/aiue.

6) Ciaeoe iaiao?aeio oa aeinoaoith oiiae iia?eueiiai c?inoaiiy
iaaaie?ii?anueei? oa?aeoa?enoeee ia?iii?oii? ooieoe?? i?e iaiaaeaiiyo ia
e?/eeuei? ooieoe?? a-oi/ie.

7) Ciaeoe iia? aneiioioe/i? oi?ioee aeey noaaa?iii?eieo ooieoe?e
ioeueiaiai ii?yaeeo oa canoinoaaoe ?o aei ciaoiaeaeaiiy ioe?iie cieco.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. An? ?acoeueoaoe aeena?oaoe?? ?
iiaeie. O aeena?oaoe?? aia?oa io?eiaii a?aeiia?aeue ia ?yae aeooaeueieo
ieoaiue uiaei ?iciiae?eo cia/aiue oa iiaiaeaeaiiy ia?iii?oieo oa
aiae?oe/ieo ooieoe?e iaoaeaeeiai c?inoaiiy. Cie?aia,

— io?eiaii iaiao?aei? oa aeinoaoi? oiiae ia eiao?oe??ioe noaiaiaaiai
?icaeiaiiy oe?eeo oa aiae?oe/ieo a e?oc? ooieoe?e aeey oiai, uia
eiaa?eoi iaeneiaeueiiai /eaia aoa iia?eueii c?inoath/ith ooieoe??th aai
aoa aea?aaeaioiee aeaye?e caaeai?e iia?eueii c?inoath/?e ooieoe??;

— ocaaaeueiaii oai?aio E?iaeaeueioa i?i ?aaoey?ia c?inoaiiy oe?ei?
ooieoe?? ne?i/aiiiai aeiaeaoiiai ii?yaeeo ia aeiaaeie aeia?eueiiai
c?inoaiiy;

— aeiaaaeaii oai?aie oeio Aae??iia oa Aae??iia-O?o/ia?oa aeey oe?eeo
ooieoe?e ioeueiaiai ii?yaeeo oa aeiaooe?a Aeyoea, e?/eeueia ooieoe?y
ioe?a yeeo ? iia?eueii c?inoath/ith. Aeey oeueiai cai?iiiiiaaii iiaee
i?aeo?ae aei aeine?aeaeaiiy aneiioioe/ieo aeanoeainoae ooieoe?e
ioeueiaiai ?iaeo, yeee aaco?oueny ia aaaaoi/eaiieo aneiioioeeao
?ioaa?ae?a;

— aaaaeaii iiaa iiiyooy ?aaoey?iiai c?inoaiiy oa “?iaeeeaoi?a” aeey
oe?eeo ooieoe?e ioeueiaiai ii?yaeeo ? o aeiaaeeo, eiee ioe?
ciaoiaeyoueny ia ne?i/aii?e nenoai? i?iiai?a, a oa?i?iao ?iciiae?eo
ioe?a anoaiiaeaii e?eoa??e oaei? ?aaoey?iino? oa iienaii oe?e? ooieoe??
?c caaeaiei “?iaeeeaoi?ii”;

— ciaeaeaii iaiao?aei? oa aeinoaoi? oiiae ia iiaiaeaeaiiy a-oi/ie
ia?iii?oieo ooieoe?e aeey oiai, uia ?o iaaaie?ii?anueea oa?aeoa?enoeea
aoea iia?eueii c?inoath/ith ooieoe??th;

— anoaiiaeaii iia? caeaaeiino? i?ae c?inoaiiyi noa?e/ii? iio?aeii?
ia?iii?oieo o?ainoeaiaeaioieo ooieoe?e oa “u?euei?noth” i?ea??anueeeo
iiiaeei aeey oaeeo ooieoe?e, a oaeiae i?ae c?inoaiiyi iaaaie?i?anueei?
oa?aeoa?enoeee, noa?e/ii? iio?aeii? oa aeiyoeiaeie iaaaie?ii?anueeeie
cia/aiiyie. Oeei ocaaaeueiaii oai?aie N.Oiii?ee, Aeae.Aiaea?niia,
O.Oaeiaia, Aeae.Eeoi?;

— io?eiai? iia? aneiioioe/i? oi?ioee aeey noaaa?iii?eieo a Rm, m? 2,
ooieoe?e ioeueiaiai ?iaeo;

— ciaeaeaii oi/i? ioe?iee aaee/ei oeio oa ieaeiueiai oeio
d-noaaa?iii?eieo a Rm, m? 2 ooieoe?e ii?yaeeo iaioiai a?ae iaeeieoe?.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. ?acoeueoaoe aeena?oaoe??
iinyoue oai?aoe/iee oa?aeoa?. Aiie iiaeooue aooe aeei?enoai? aeey
?icaeoeo oai??? oe?eeo oa ia?iii?oieo ooieoe?e a ??cieo iai?yieao, a
oaeiae a oai??? iioaioe?aeo.

Ne?ae caoaaaeeoe, ui aeaye? ?acoeueoaoe aeena?oaoe?? ocaaaeueithaaeenue
oa aeei?enoiaoaaeenue ?ioeie aaoi?aie. Oae, N.TH.Oaai?ia [6] aeacaa
oe?oee aeiionoeiee eean noaaa?iii?eieo a Rm (m? 3 ) ooieoe?e, i?ae
aeacaiee a aeena?oaoe?? eean iie?noaaa?iii?eieo ooieoe?e ioeueiaiai
ii?yaeeo. ss.ss.I?eooea aeei?enoaa iaoiaeeeo noaoo? [19] aeey ?ioiai
ocaaaeueiaiiy oai?aie A.E?iaeaeueioa.

Iniaenoee aianie caeiaoaa/a. An? iaoeia? ?acoeueoaoe, aeeth/ai? a
aeena?oaoe?th, iaea?aeai? caeiaoaa/ai iniaenoi. O ni?euei?e noaoo? [3] c
iaeiei ?c iaoeiaeo eiinoeueoaio?a – A.A.Aieueaeaa?aii — ni?aaaoi?ia?
iaeaaeeoue iinoaiiaea caaea/? ? aeaye? canoinoaaiiy io?eiaieo
?acoeueoao?a.

Iniiai? ?acoeueoaoe o?aoueiai ?icae?eo aeena?oaoe?? iioae?eiaai? a
ni?eueieo c iaoeiaei eiinoeueoaioii I.I.Oa?aiaoith noaooyo [18-20]. O
oeeo i?aoeyo ni?aaaoi?ia? iaeaaeeoue eaia 1 c [19] oa caoaaaeaiiy, ui
aea/aiiy iia?eueiiai c?inoaiiy eiaa?eoia iaeneiaeueiiai /eaio ?yaeo
Oaeei?a iiaea aeyaeoenue aecia/aeueiei i?e iienaii? iia?eueiiai
c?inoaiiy ?aooe caaaeaieo aeua oa?aeoa?enoee aiae?oe/ieo ooieoe?e. ?c
noaooae [2, 6, 24] a aeena?oaoe?th oa?eoee o?eueee ?acoeueoaoe aaoi?a.
Ia?aoo?, ?acoeueoaoe noaoo? [1] ? /anoeiaei aeiaaeeii oai?aie 3.1 c
aeena?oaoe??.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. An? ?acoeueoaoe aeena?oaoe?? c
aeiaaaeaiiyi aeiiia?aeaeenue ia i?nueeiio nai?ia?? c oai??? aiae?oe/ieo
ooieoe?e o Eueaia? (ea??aieee A.A.Aieueaeaa?a, A.A.Eiiae?aothe,
I.A.Neane?a). Iniiai? ?acoeueoaoe aeena?oaoe?? aeiiia?aeaeenue ia VI oa
VII eiioa?aioe?yo c eiiieaeniiai aiae?co oa aeeoa?aioe?aeueieo ??aiyiue
(i.*i?iiaieiaea, 1987 ?. oa 1989 ?.), ia ?anioae?eainuee?e
oeie?-nai?ia?? c oai??? oe?eeo oa noaaa?iii?eieo ooieoe?e (i.Oa?e?a,
1990 ?.), ia i?aeia?iaeieo eiioa?aioe?yo, i?enay/aieo 100-??//th c aeiy
ia?iaeaeaiiy N.Aaiaoa (i.Euea?a, 1992 ?.), iai’yo? Aaina Aaia
(i.*a?i?aoe?, 1994 ?.), “Ooieoe?iiaeuei? i?inoi?e aeeoa?aioe?aeueieo
iia?aoi??a ? iae?i?eiee aiae?c” (i.I?aaa, 1995 ?.) oa “No/ani? i?iaeaie
iaoaiaoeee” (i.*a?i?aoe?, 1998 ?.), ia 5-?e
o?inueei-iieuenueei-oe?a?inuee?e ni?euei?e oeie? c eiiieaeniiai aiae?co
(i.Ethae?i, 1996 ?.), ia nai?ia?ao c oai??? ooieoe?e eiiieaenii? ci?iii?
a ?inoia?-ia-Aeiio (ea??aiee TH.O.Ei?iaaei?e), Oa?eia? (ea??aieee
A.ss.Eaa?i, E.A.Ino?ianueeee, A.I.A?eoei), Ee?a? (ea??aiee
I.I.Oai?acia), ia Euea?anueeiio ?aa?iiaeueiiio nai?ia?? c iaoaiaoe/iiai
aiae?co (ea??aiee I.I.Oa?aiaoa).

Ioae?eaoe??. ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a 26 ?iaioao (17 aac
ni?aaaoi??a), c yeeo 23 aeo?iaeuei? noaoo? (15 aac ni?aaaoi??a) o
aeaeaiiyo ?c ia?ae?e?a, caoaa?aeaeaieo AAE Oe?a?ie, oa 3 o iaoa??aeao
i?aeia?iaeieo iaoeiaeo iaoaiaoe/ieo eiioa?aioe?e.

No?oeoo?a ? ia’?i ?iaioe. Aeena?oaoe?y neeaaea?oueny ?c ia?ae?eo
oiiaieo iicia/aiue, anooio, oanoe ?icae?e?a, ?icaeoeo ia i?ae?icae?ee,
aeniiae?a ? nieneo aeei?enoaieo aeaea?ae. Ianya aeena?oaoe?? 285
noi??iie. Nienie aeei?enoaieo aeaea?ae caeia? 13 noi??iie ? aeeth/a? 116
iaeiaioaaiue.

Aaoi? aeneiaeth? ue?o iiaeyeo nai?i iaoeiaei eiinoeueoaioai i?io.
A.A.Aieueaeaa?ao oa i?io. I.I.Oa?aiao?.

CI?NO ?IAIOE

O anooi? iaa?oioiao?oueny aeooaeuei?noue oaie, iiaea?oueny iaoa ?
caaea/? aeine?aeaeaiiy, iaoeiaa iiaecia, i?aeoe/ia cia/aiiy ? ai?iaaoe?y
io?eiaieo ?acoeueoao?a, e?euee?noue ioae?eaoe?e.

O ia?oiio ?icae?e? iaaiaeyoueny iniiai? icia/aiiy oai??? ?.Iaaaie?iie,
aea?oueny iaeyae e?oa?aoo?e ca oaiith aeena?oaoe?eii? ?iaioe c aeacaiiyi
ia?ica’ycaieo i?iaeai.

O ae?oaiio ?icae?e? iaaaaeaii oi?ioethaaiiy iniiaieo ?acoeueoao?a
?iaioe, ye? aeathoue a?aeiia?aeue ia i?iaeaie iinoaaeai? o ia?oiio
?icae?e?.

Iaoae

O aeiaaeeo 00 (D(a)>0), oi a
iaceaathoue aeiyoeiaei cia/aiiyi a ?icoi?ii? Iaaaie?iie aai
iaaaie?ii?anueeei aeiyoeiaei cia/aiiyi (a ?icoi?ii? Aae??iia aai
aae??ii?anueeei aeiyoeiaei cia/aiiyi).

I?ae iia?eueii c?inoath/ith a oi/oe? +Y (aai i?inoi iia?eueii
c?inoath/ith) ooieoe??th aoaeaii ?icoi?oe iaia?a?aio iaa?ae’?iio
c?inoath/o aei +Y ia [0,+Y) ooieoe?th l oaeo, ui l(2x)~l(x), x® +Y.
Ooieoe?y l(x) iaceaa?oueny iia?eueii c?inoath/ith a oi/oe? 1 ce?aa (a
e?aiio ieie? oi/ee 1), yeui ooieoe?y h(x)=l(1-1/x) ? iia?eueii
c?inoath/ith a oi/oe? +Y, oiaoi l((1+x)/2) ~ l(x), x® 1-, ? l –
iaia?a?aia iaa?ae’?iia c?inoath/a aei +Y ia [0,1) ooieoe?y.

A o?aoueiio ?icae?e? aeena?oaoe?? ?icaeyaeathoueny aea? caaea/?, ye?
iieyaathoue a ciaoiaeaeaii? iaiao?aeieo oa aeinoaoi?o oiia ia
eiao?oe??ioe an oa ioe? zk aiae?oe/ii? ooieoe?? f, caaeaii? noaiaiaaei
?yaeii (1) aeey oiai, uia iniiai? ?? oa?aeoa?enoeee lnM(r,f), lnm(r,f),
T(r,f) oa N(r,1/f) aoee, ii-ia?oa, iia?eueii c?inoath/eie ooieoe?yie a
oi/oe? +Y aai a oi/oe? 1 ce?aa, a ii-ae?oaa, aea?aaeaioi? aeaye?e
caaeai?e iia?eueii c?inoath/?e ooieoe??.

A i?ae?icae?e? 3.2 iieacaii, ui T(r,f), lnM(r,f) ? lnm(r,f) o aeiaaeeo
oe?ei? ooieoe?? ? iia?eueii c?inoath/eie iaeii/anii, i?e/iio iaiao?aeii
? aea?aaeaioieie. (sseui f ia? ioeueiaee ii?yaeie, oi oaeith aoaea ?
ooieoe?y N(r,1/f)). Ioaea, i?iaeaia iia?eueiiai c?inoaiiy iniiaieo
oa?aeoa?enoee o aeiaaeeo oe?ei? ooieoe?? caiaeeoueny aei aea/aiiy
iia?eueii? ci?ie eiaa?eoio iaeneiaeueiiai /eaia ?? noaiaiaaiai
?icaeiaiiy, a ?ica’yceii inoaiiuei? ?

Aeey oiai uia lnm(r,f) aoea iia?eueii c?inoath/ith ooieoe??th,
iaiao?aeii ? aeineoue, uia ?nioaaea c?inoath/a iine?aeiai?noue (nk)
iaoo?aeueieo /enae oaea, ui

?

Iane?aeie 3.5. sseui eiao?oe??ioe oe?ei? ooieoe?? (1) oae?, ui
|an|2/|an-1an+1|?1, oi iaiao?aeiith oa aeinoaoiueith oiiaith iia?eueiiai
c?inoaiiy ln m(r,f) ? aeeiiaiiy ni?aa?aeiioaiiy

Oai?aia 3.4. Aeey oiai uia ooieoe?y N(r,1/f) aoea iia?eueii
c?inoath/ith, iaiao?aeii ? aeineoue, uia aeeiioaaeanue oiiaa

aea (zn) – caioia?iaaia a ii?yaeeo ianiaaeaiiy iiaeoe?a iine?aeiai?noue
ioe?a ooieoe?? f.

Iaoae a – iaa?ae’?iia iaia?a?aia c?inoath/a aei +Y ia [0,+Y) ooieoe?y
oaea, ui xa(x) – iioeea ooieoe?y, a Ba – eean oe?eeo o?ainoeaiaeaioieo
ooieoe?e (1) oaeeo, ui

ln |an|F -na(n), n? n0.

I?e?iaeii aeieea? caieoaiiy, i?e yeeo oiiaao ia ooieoe?th a aeey eiaeii?
fI Ba ooieoe?y ln M(r,f) ? iia?eueii c?inoath/ith.

Aeei?enoiaoth/e oai?aio 3.3, ie io?eio?ii oaea oaa?aeaeaiiy.

Oai?aia 3.5. Aeey oiai uia ln M(r,f) aoea iia?eueii c?inoath/a aeey
eiaeii? ooieoe?? fI Ba iaiao?aeii ? aeineoue, uia

Iaoae oaia? f — aiae?oe/ia a iaeeie/iiio e?oc?. Caoaaaeeii, ui yeui
m(r)=O(1), r®1- , oi, caaaeyee ianiaaeaiith, m(r) ? iia?eueii
c?inoath/ith a oi/oe? 1 ce?aa ooieoe??th. Oiiaa m(r)=O(1), r® 1- ,
aeeiio?oueny oiae? ? o?eueee oiae?, eiee |an|=O(1), n®+Y . Oiio
iaaeae? aoaeaii aaaaeaoe, ui

A i?ae?icae?e? 3.3 io?eiaii iiaiee aiaeia oai?aie 3.3 aeey aiae?oe/ieo a
e?oc? ooieoe?e (oai?aia 3.9), c yei?, cie?aia, aeieeaathoue ianooii?
iane?aeee.

Iane?aeie 3.6. sseui eiao?oe??ioe aiae?oe/ii? a { z : |z|0 .

Ine?eueee aeey aeineoue oe?ieeo eean?a ?yae?a Ae???oea ln M(s,F)~ ln
m(s,F) i?e s ® A , oi i?iaeaia anoaiiaeaiiy oiia, ye? caaacia/othoue
aeaiaaeaioi?noue ln M(s,F)~ F(s) i?e s®A caiaeeoueny aei iiae?aii?
i?iaeaie noiniaii aeaiaaeaioiinoi ln m(s,F)~ F(s) i?e s ® A .

Oai?aia 3.12. Iaoae F I W (A) . Aeey oiai uia ln m(s,F)~ F(s) i?e s
® A , iaiaoiaeii i aeineoue, uia aeey eiaeiiai e >0 :

1) inioaaei n0 = n0 (e) I N oaea, ui aeey anio n? n0

2) inioaaea c?inoath/a iineiaeiaiinoue (nk) iaoo?aeueieo /enae oaea,
ui

i

(5) 1, aea v(t) –
iia?eueii c?inoath/a a oi/oe? 1 ce?aa ooieoe?y. Iai?eeeaae, oai?aia
oeio Aae??iia ia? aeaeyae.

Oai?aia 4.5. Iaoae v(r) iia?eueii c?inoath/a a oi/oe? 1 ooieoe?y
oaea, ui aeua ?acoeueoaoe iathoue e?eoa??aeueiee oa?aeoa?.

II. Aeey oe?eeo ooieoe?e ioeueiaiai ii?yaeeo ? aeey aeiaooe?a Aeyoea,
e?/eeueia ooieoe?y ioe?a yeeo ? iia?eueii c?inoath/ith, aeiaaaeai?
oai?aie oeio Aae??iia oa oeio Aae??iia-O?o/ia?oa.

III. Aaaaeaii iiaa iiiyooy ?aaoey?iiai c?inoaiiy oa “?iaeeeaoi?a” aeey
oe?eeo4 investigating asymptotical properties of entire functio

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