ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri Istitute of Techology, Post-Suka, Siliguri, Dist-Dareelig, Pi- 734009, West egal, Idia (Received March 3, 0, acceted Aril 0, 0 Abstract This aer is cocered with the study of the axiu odulus ad the coefficiets of the ower series exasio of a fuctio of several colex variables aalytic i the uit olydisc Keywords: Aalytic fuctio, order, lower order, several colex variables, uit olydisc Itroductio ad Defiitios Let f of f ( z o z r ( z c z be aalytic i the uit disc { z : z < } 0 U ad M ( r M ( r, f be the axiu I 968 Sos [8] itroduced the followig defiitio of the order ad the lower order as su log log M ( r, f li r if log ( r Maclae [6] ad Kaoor [5] roved the followig results which are the characterizatio of order ad lower order of a fuctio f aalytic iu, i ters of the coefficiets c Theore [6] Let f ( z c z be aalytic iu, havig order ( 0 0 log log li su + log c The Theore [5] Let f ( z c z be aalytic iu, havig lower order ( 0 0 log log li if + log [0] [0] Notatio 3 [7] log x x, ex x x ad for ositive iteger, c The [ ] [ ] [ ] [ ] log x log (log x, ex x ex(ex x I a aer [4] Juea ad Kaoor itroduced the defiitio of -th order ad lower -th order ad i 005 aeree [] geeralized Theore ad Theore for -th order ad lower -th order resectively Defiitio 4 [4] If defied as f ( z c z be aalytic iu, its -th order 0 ad lower -th order are + Corresodig author E-ail address: rata_38@yahooco Published by World Acadeic Press, World Acadeic Uio
96 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc [ ] su log ( M r li r, if log ( r Theore 5 [] Let f ( z c z be aalytic i U ad havig -th order ( 0 0 + li su + [ ] log log c The Theore 6 [] Let f ( z c z be aalytic i U ad havig lower -th order ( 0 0 + li if + [ ] log log c The I 008 aeree ad Dutta [] itroduced the followig defiitio Defiitio 7 Let f ( z, z be a o-costat aalytic fuctio of two colex variables z ad z holoorhic i the closed uit olydisc P:( z, z : z ;, the order of f is deoted by ad is defied by { } if { μ 0 : (, ex > F r r < r r ; for all 0 < r0 ( μ < r, r < } Equivalet forula for is log log F( r, r lisu r, r log( r ( r I a reset aer [3] aeree ad Dutta itroduce the defiitio of -th order ad lower -th order of fuctios of two colex variables aalytic i the uit olydisc ad geeralized the above results for fuctios of two colex variables aalytic i the uit olydisc Defiitio 8 Let f ( z, z c z z be a fuctio of two colex variables z, z holoorhic i the uit olydisc ad, 0 U {( z, z : z ;, } Fr (, r ax{ f( z, z: z r;,}, be its axiu odulus The the -th order ad lower -th order are defied as [ ] su log Fr (, r li r, r, if log( r( r Whe, Defiitio 8 coicides with Defiitio 7 Theore 9 Let f ( z, z be aalytic i U ad havig -th order ( 0 The + [ ] log c li su, + log Theore 0 Let f ( z, z be aalytic i U ad havig lower -th order ( 0 The + [ ] log c li if, + log μ JIC eail for cotributio: editor@icorguk
Joural of Iforatio ad Coutig Sciece, Vol 6 (0 No 3, 95-06 97 Whe the fro Theore 9 ad Theore 0 we get this two theores Theore Let f ( z, z be aalytic i U ad havig order ( 0 The log log c li su, + log Theore Let f ( z, z be aalytic i U ad havig lower order ( 0 The log log c li if, + log I this aer we cosider a ore geeral situatio i the case of aalytic fuctios of several colex variables i the uit olydisc ad for which we itroduce the followig defiitio Defiitio 3 Let f ( z, z, z c z z z be a fuctio of colex,, 0 variables z, z, z holoorhic i the uit olydisc U {( z, z, z : z ;,, } ad Frr (,, r ax{ f( z, z, z : z r;,, }, be its axiu odulus The the order ad lower order are defied as su log log Frr (,, r li r, r, r if log( r( r( r Whe, Defiitio 3 coicides with Defiitio 7 I this aer we fid a siilar aalytic exressio for ad i ters of the coefficiets c several colex variables olydisc Here f ( z, z, z c z z z will deote a fuctio aalytic i the uit Leas,, 0 The followig leas will be eeded i the sequel Lea Let the axiu odulus Frr (,, r of a fuctio f ( z, z, z aalytic iu, satisfy log Frr (,, r < A ( r ( < A, < for all r such that r 0 ( A, < r < ;,,, The for all > ( A, > ;,, 0 log c S( A, + for where JIC eail for subscritio: ublishig@wauorguk
98 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Proof At first defie sequeces { r } by Also, The r as A S A, ( + + ( + ( r ;,, A for all,, + + + f (0, 0,0 z z z c (! f ( z, z z dz dz dz ( π i z z z F ( r, r r r r r Frr (, r r Fro ( ad ( we have for all z r z r z r > ( A, > ;,, 0 + + + ( log c log F( r, r r log r ( ( < A r + r [ + O(] + + A ( A A + [ + O(] + + A A + [ (] + O A + + A + A [ + O(] A A JIC eail for cotributio: editor@icorguk
Joural of Iforatio ad Coutig Sciece, Vol 6 (0 No 3, 95-06 99 Therefore where This roves the lea + [ + { + O(}] A + + A (+ A log c S( A, + A S A, ( + + ( Lea Let f ( z, z, z be aalytic iu ad satisfy D c ex (, 0, 0 < C < C < < D< for all > ( C, D;,, The for all r such that r0 ( C, D < r < ;,, 0 where Proof For all log Frr (,, r < TCD (, log r D D D D D, TCD (, C D [ + o(] > ( C, D;,,, 0 Now for z r < ;,, D c ex ( < C D ( C ex JIC eail for subscritio: ublishig@wauorguk
00 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc D where D Choose <,, 0 Frr (,, r c r r r < K + C r D (, ( 0, ex 0 0 0 + 0 + : + 0 +,, ex 0 0 0 + 0 K( + C r, + C M M( r ;,, log r where [x] deotes the greatest iteger ot greater tha x Clearly M( r as r for all,, The above estiate of Frr (,, r for r;,, sufficietly close to gives, where for if M +, the Frr (,, r K (,, Mr ( Hr ( r / < 0 0 + 0 + M + + H( r ax ex C r ;,, (3 for all,, M + C > log r + ie C < log r + ie exc r < r Therefore the ifiite series r i (3 is bouded by r Sice > 0, we have M + + r for all,, JIC eail for cotributio: editor@icorguk
Joural of Iforatio ad Coutig Sciece, Vol 6 (0 No 3, 95-06 0 M + C log log r < log log( r + log r r + log r r as r Thus for r sufficietly close to, + for all,, r M + o( by + The axiu of exc r assue at the oit + log Hr ( C + log r C ( log + r + ( C log C C r + + ( + log ( + log r r + C ( + log r Thus for r ;,, sufficietly close to, fro (3 Frr (,, r Mr ( Hr ( o( + (,, 0 0 0 < + + M( r H( r + o( M( r H( r + o( [ + O(] K ad H ( r is give (4 JIC eail for subscritio: ublishig@wauorguk
0 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Therefore where { } log Frr (,, r < log Mr ( + log Hr ( + O( This roves the lea 3 Theores + [] C ( + log + ( [ (4] + O fro r ( + log r + C O( + ( + log r + C log ( + r + O( log [ O(] + C + + r D D D D D C D [ + o(] log r log Frr (,, r < TCD (, log r D D D TCD (, C D [ + o(] We rove the followig theores Theore 3 Let f ( z, z, z be aalytic i U ad havig order (0 The log log c li su (5 + log Proof If c is bouded by K for all ;, the c z z z is bouded by,, 0 Therefore K ( r D D JIC eail for cotributio: editor@icorguk
Joural of Iforatio ad Coutig Sciece, Vol 6 (0 No 3, 95-06 03,, 0 Frr (, r c r r r ( r < ex ( r for ay 0< ε < ad for all r ;,, sufficietly close to K Therefore log log Frr (, r li sur, r, r ε log ( r sice 0< ε < arbitrary, 0 ad so (5 is satisfied Thus we eed to cosider oly the case li su c I this case all the log + i (5 ay be relaced by log First let 0 < <ad > The for all r ;,, sufficietly close to, ε Usig Lea with > ( ;, 0 log Frr (, r < ( r A, it follows fro the above iequality that for + + + log c ( Therefore log log c li su,, + log Sice > is arbitrary, it follows that log log c li su,, (6 + log Sice f ( z, z, z is aalytic iu, the above iequality is trivially true if ad the right had side is iterreted as i this case Coversely, if JIC eail for subscritio: ublishig@wauorguk
04 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc θ li su,, the 0 θ First let θ < ad choose θ < θ < The for all sufficietly large ;,,, 0 log log c log log c < Usig Lea with C, D θ it follows fro the above iequality that for all r such that r r ( θ < < ;,,, that is θ θ θ log Frr (,, r < θ log [ + o(] r θ θ log log Frr (,, r < log( + log log + log[ + o(] θ θ θ θ r log log log log Frr (,, r θ r li sur, r, r li su,, r r r θ log ( r log ( r θ < θ Sice θ > 0 is arbitrary, it follows that log log c θ li su,, + log If θ, the above iequality is obviously true Iequality (6 ad (7 together gives (5 whe li su c θ This roves the theore Theore 3 Let f ( z, z, z be aalytic i U ad havig lower order (0 The Proof Let First suose that 0< A < log log c li if,, + log li if,, log log c A log (7 (8 JIC eail for cotributio: editor@icorguk
Joural of Iforatio ad Coutig Sciece, Vol 6 (0 No 3, 95-06 05 Fro (8, for 0< ε < A <, log c > for > M M ( ε ;,, Also Choose The fro (9 where k is a suitable costat Frr (,, r c r log Frr (,, r log c + log r (9 log where, r log Frr (,, r > log log log r r r log log r r > log k r A ε log log Frr (,, r > log log + O( A ε r log log log log Frr (,, r A ε r > + O( A ε log ( r log ( r log log Fr (, r, r li ifr, r, r log ( r A ε A + ε Sice 0< ε < A < is arbitrary, A A This ilies JIC eail for subscritio: ublishig@wauorguk
06 Rata Kuar Dutta, et al: O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc A + This iequality holds obviously whe A 0 For A the above arguets with a uber K arbitrarily ear to i lace of A ε, give This roves the theore 4 Referece + [] D aeree O -th order of a fuctio aalytic i the uit disc Proc Nat Acad Sci Idia 005, 75(A, IV: 49-53 [] D aeree ad R K Dutta Relative order of fuctios of two colex variables aalytic i the uit disc J Math 008, : 37-44 [3] D aeree ad R K Dutta O -th order of a fuctio of two colex variables aalytic i the uit olydisc Proc Nat Acad Sci Idia i ress [4] O P Juea ad G P Kaoor Aalytic Fuctios-Growth Asects Pita Advaced Publishig Progra, 985 [5] G P Kaoor O the lower order of fuctios aalytic i the uit disc Math Jao 97, 7: 49-54 [6] G R Maclae Asytotic Value of Holoorhic Fuctios Rice Uiversity Studies, Housto, 963 [7] D Sato O the rate of growth of etire fuctios of fast growth ull Aer Math Soc 963, 69: 4-44 [8] L R Sos Regularity of growth ad gas J Math Aal Al 968, 4: 96-306 JIC eail for cotributio: editor@icorguk