On The Generalized Type and Generalized Lower Type of Entire Function in Several Complex Variables With Index Pair (p, q)

Transcription

1 O he eeralized ye ad eeralized Lower ye of Eire Fucio i Several Comlex Variables Wih Idex Pair, Aima Abdali Jaffar*, Mushaq Shakir A Hussei Dearme of Mahemaics, College of sciece, Al-Musasiriyah Uiversiy, Baghdad, Iraq Absrac I he rese aer, we will sudy he geeralized, -ye ad geeralized lower, -ye of a eire fucio i several comlex variables wih resec o he roximae order wih idex air, are defied ad heir coefficie characerizaios are obaied Keywords: Eire fucio, geeralized ye, geeralized lower ye,idex air حول اعمام النوع واعمام النوع االدنى لدالة كمية ذات متغي ارت معقدة متعددة مع دليل الزوج, أيمن عبد عمي جعفر* مشتاق شاكر عبد الحسين قسم الرياضيات كمية العموم الجامعة المستنصرية بغداد الع ارق الخالصة: في بحثنا هذا سوف ندرس اعمام النوع q, واعمام النوع االدنى q, لدالة كمية ذات متغي ارت معقدة متعددة بالنسبة الى تقريب الرتبة لدليل الزوج q, من خالل تعريفها عمى المعامالت المميزة Iroducio Kumar ad ua ] le f z, z2,, z be a eire fucio z z, z2,, z C Le be a regio i osiive hyer oca ad le C deoe he regio obaied from by a similariy rasformaio abou he origi, wih raio of similiude Le d su z, where regio Le f z z f z, z 2 * aima_mah20@yahoocom z 942 z z 2 z, ad le deoe he boudary of he, 2,, 0 2,, z a z z a z, 2, be he ower series exasio of he fucio f z Le M max f 0 z z o characerize he growh of f, order ad ye of f are defied as

2 lim su, ad lim su For 0, he maximum erm of eire fucio f z is defied as see ad 3] max{ a 0 For eire fucio d f z 0 } a z erms of he coefficies of is aylor exasio as, AA ol'dberg 4,h] obaied he order ad ye i lim su a ad e where lim su{ a d d ] max r ; r r r2 2 r r },0 he coce of, -order, lower order, -order,, -ye ad lower, -ye of a eire fucio f z, z,, z havig a idex air,, was iroduced by Jueja 2 e al 5], 6] hus f z is said o be of, -order ad lower, -order if ] su lim if,, where ad q are iegers such ha q If b,, where b, if q ad b 0 if q, he he, ] su lim, if ad m] x ex m],, x -ye ad lower, m] x exex m ] 0] 0] 0 x wih x ex x x m] -ye is give by x, m 0,, 2, rovided ha he growh of a fucio f z ca be sudied i erms of is order ad ye, bu hese coces are iadequae o comare he growh of hose fucios which are of he same order ad of ifiie ye Hece, for a refieme of he above growh scale, oe may uilize roximae order he coce of which is 7] as follows: A fucio defied o 0, is said o be a roximae order of a eire fucio wih idex air, if i saisfies he roeries: lim ad lim ] 0, where q Now, we defie he geeralized, -ye ad geeralized lower, -ye f z wih resec o a give roximae order as of 943

3 ] su lim,0 2 if A roximae order is called a roximae order of a eire fucio f z wih idex, if is o-zero ad fiie ad he fucio f z is said o be of erfecly regular, growh wih resec o is roximae order if I he rese aer we obai coefficie characerizaios of geeralized, -ye ad geeralized lower, -ye By 7] q ] of he eire fucio f z is a moooically icreasig fucio of for 0, so we defie a sigle valued real fucio k of k for k k0 such ha ] ] k q A q k 3 he we have he followig : Lemma Le be a roximae order wih idex air, ad le k be defied as i 3 he d k lim 4 k d k A ad for every η wih o k A lim 5 k k where A = whe, 2,2 = 0 oherwise Proof d k d d k d{ A } A ] assig o he limis Agai, k A, k akig limis we ge 5 Lemma 2 Le k we obai 4 f z a z be a eire fucio havig roximae order wih idex air, Le ad be he geeralized, -ye ad geeralized lower, -ye of f z wih resec o a roximae order he 0 su lim if ] 6 Proof: By he maximum erm i 8] ad by usig he ye ad lower ye 6], we have 944

4 For 0, he maximum erm of eire fucio f z is defied as ad su f lim if, f max{ a, he from 6], we ge 6 2 Mai esul heorem 2 If 0 } f z a z is a eire fucio wih roximae order ad, -order wih idex air,, he he geeralized, -ye of f z wih resec o he roximae order is give by A { } M lim su { a d }, 2 where if, 2,2 M e if, 2, if for all oher idex air, ad 2 2 ;, 2,,, for, 2, ;, 2,,, for 2 q 0 ; a leas oe, 2,, 0 Proof From 6 for every 0 ad for all ex { }, for all such ha 0 0, a d ex { } 22 Now choose such ha A 23 g For, 2,2, 23 is reduced o, which gives ha k ad 945

5 946 Usig he resuls 22 yields o d a q Passig o limis, we have usig 5 3 } { su lim ] 2 d a q 24 For 2,2, q, he equaio 23 becomes ], which imlies ha k ad Hece, 22 is wrie as } { d a, where ad Sice lim sice ε is very small ad lim so su lim d a 25 Agai, for 2,, q, 23 is reduced o which gives k k Equaio 22 is covered io

6 a d e Passig o limis we have lim su e a d 26 Equaios 24, 25 ad 26 combie io lim su q { a d } A M 27 o rove he reverse iequaliy, le A lim su M q { a d } For ay 0, we have for all m0 m0 q M a d ex ex, where So, q M max ex 28 0 For, 2, ad 2, 2, usig 4 i ca be easily see ha he maximum value o he righ-had side is aaied for ex q hus, for sufficiely large we ge from 28 q ] o Proceedig o limis 29 Cosider whe, 2, Le e M, equaio 28 is he reduced o 947

7 e M ad assig o limis we ge 20 If, 2,2, i order o ge he maximum value of he righ-had side of he iequaliy 28 is give by M which reduces 28 o, M O akig limis we ge 2 29, 20 ad 2 give Sice his iequaliy holds for every 0, so his ad 27 ogeher rove he heorem A akig ad k k, we have he followig corollary which gives a formula for he, -ye of he eire fucio f z heorem 22 Le 0 ad, -order such ha f z a z be a eire fucio havig he roximae order a a, forms a o-decreasig fucio of for m0 he he geeralized lower, of f z is give by where M, A ad are he same as give i heorem 2 Proof Sice by hyohesis, -ye is a o-decreasig fucio of for m0 We have for ifiiely may values of ; oherwise f z ceases o be a eire fucio So Whe as, he erm a z becomes maximum ad he a, v for Firs, le 0, i view of Lemma 2, for ay ε saisfyig we ge ad for all 948

8 ex ] 22 m m2 Le a ad a z m m, m m z m be wo cosecuive maximum erms of z of for m0, we have for m m2, m f he sice m0 m 2 23 Ad m2 a am for 2 Hece, 22,23 ad 24 give 24 ex ] a d or, X { ex A { g } A a d }] { } ex A { 25 We oe ha he miimum value of he fucio S is a o-decreasig fucio A ex { }}] { } A ex A { ex { is aaied a a oi 0 saisfyig E { } 26 For, 2,, 26 gives Hece X mi S mi 0 ex { e ] }] e 27 For, 2,2, 26 becomes Hece, mi S { } 0 28 }}] 949

9 For, 2,2 ad 2,, 26 is reduced o So mi S 0 { ex{ } e 29 } 25,27,2,8 ad 29 combie io lim if X M 220 he iequaliy 220 is obvious if 0 Whe, above argumes wih a arbirarily large umber i lace of leads o lim if X We ow rove ha sric iequaliy cao hold i 220 for if i holds, he here exiss a umber such ha M lim if q { a d } A Le be such ha, he for all m0 a d ex q M A herefore, for sufficiely large ad we have q f ex A M, 22 For, 2,, choose ], he i view of Lemma, e or, 950

10 Passig o limis 222 I case, 2,2, choose { he 22 is reduced o M } k, or, M ] which gives o assig o limis 223 Furher, cosider, 2, ad 2, 2 if is give by he e e e ; or, ex ex ] M q q o ex q Proceedig o limis we have 224 So 222,223 ad 224 are formed i o which is a coradicio Hece he roof of he heorem is comlee 95

11 Corollary 23 Le 0 lower, -ye 0 m 0, he f z a z be a eire fucio havig he, q -order ad such ha is o-decreasig fucio of for M lim if q2 ] { a } A Ackowledgme he auhor is hakful o he referees for heir helful commes ad suggesios for imrovig he aer efereces Kumar D ad ua Deei, 20, O he aroximaio of eire fucio of several comlex variables, Ieraioal Mahemaical Forum, 6, : oala J Krisha, 969, Maximum erm of a ower series i oe ad several comlex variables, Pacific J Mah 29 : oala J Krisha, 970, Probabilisic echiques leadig o a Valiro-ye heorem i several comlex variables, A Mah Sais 4, : ol'dberg AA, 959, Elemeary remarks o he formulas for defiig order ad ye of fucios of several variables, Akad Nauk Armja SS Dokl, 29, : Jueja OP, Kaoor P, ad Bajai SK, 976, O he, -order ad lower, - order of a eire fucio J eie Agew Mah, 282, : Jueja OP, Kaoor P, ad Bajai SK, 977, O he, -ye ad lower, -ye of a eire fucio J eie Agew Mah 290, : Nada, Krisha, Doherey P ad Srivasava SL, 980, Proximae order of a eire fucio wih idex air, Idia J ure al Mah,, : Susheel Kumar ad Srivasava S, 20, Maximum erm ad lower order of eire fucio of several comlex variables, Bullei of Mahemaical aalysis ad Alicaio, 3, :