Issue, Volue, 7 5 Growth of a Class of Plursubharoc Fucto a Ut Polydsc I AITASU SINHA Abstract The Growth of a o- costat aalytc fucto of several coplex varables s a very classcal cocept, but for a fte doa t s a recet cocept tated by Jueja ad Kapoor[],ad later o substatated by Sha[]. A Ut Polydsc s the ost fudaetal exaple of a Copact ea Surface, there s a upsurge the area as beg reflected [4,5]. Ths s a very portat cocept ad ts applcatos ca also be see Coplex Aalytcal yacs. However ths preset artcle we shall be cocetratg o the growth paraeter type for such fuctos ad also descrbe ts Geoetrcal Propertes.We have vestgated upo soe fer results o the growth of slowly varyg fuctos. The above cocept of Growth ca also be utlzed Coputer aded Toography[7]. Keywords Aalytc Fucto, Coplex Aalytcal yacs,growth Paraeter, Type.. INTOUCTION I ths secto we frst defe a class E (β where β L. The class of such fuctos L was defed by Jueja Kapoor[] ad Sha []. Such classes of fuctos were tated by Sereeta [6] ad the has bee used extesvely by Sha[3]. To study the fuctos havg fast or slow growths a portat cocept of (p,q order ad type was studed by ay authors the past. I the preset paper we have troduced the cocept of (q, order ad type for Fuctos havg fast growth. The above results ca also be used Coputer Aded Toography as, by the faous ea- appg Theore ay sply coected doa ca be coforally apped oto Ut- sc,so we ca project the three desoal Tuour a two d.plae ad study ts growth through the above ethods ad the we ca apply verse trasfor to study ts Growth[7]. Wor ths drecto s progress. Let C be a doa. efto : Let E (β be a class of fuctos ϕ ( satsfyg the followg Propertes,. ϕ ( s upper se-cotuous o.. ϕ ( s ootoe, odecreasg each of the varables,,.... auscrpt receved Jauary 9, 7: evsed verso receved Oct. 4, 7. 3. ϕ ( s plurcovex the varables, β (log(, β (log(,... β (log(, that s to say for every t t,... t ad s ( s, s,... s I, ad for all λ, μ wth λ + μ exp ϕ exp λϕ ( t, t { β ( λβ (log( t + μ ( β (log( s } { β ( λβ (log( t + μ ( β (log( s },... t + μϕ ( s, s,... s (.. (. Upo substtutg β ( x x, the detty fucto the class reduces to the class of fuctos ϕ ( defed by Jueja ad Kapoor[]. efto : Let ϕ axϕ(, < t < be the Here I {,... :,,,... } axu odulus of the fucto ϕ (, ad s the Ut Polydsc. The geeralzed order for ϕ ( s defed as, α( (, ( lsup t ϕ ϕ t β (log( t here α Δ ad β L. (. Exaple: ϕ (, α [ β (log( β (log( ].It ca be easly see that the above fucto has geeralzed order. efto 3: Let α Δ ad < < be the geeralzed type of f (Z s defed as, E (β f, lsup α log( (.3 [ ] where sup f ( ; Here Z belogs to. f, Z Z belogs to f the pot z z z,,...,..., INTENATIONAL JOUNAL OF APPLIE ATHEATICS AN INFOATICS
Issue, Volue, 7 5 efto 4: For α Δ ad β L, the geeralzed type wth respect to oe of the varables eepg the others fxed s defed as f, lsup (.4 α [ β (log( as, + α( ϕ lsup (.4. β (log( efto 5: Let α Δ ad β L ad let Theore.: The geeralzed order ca be obtaed fro the efto of Geeralzed Type. ϕ β be a fucto of fte geeralzed type. Let Proof: Fro the defto of lt superor we have for ay ( a, a,... a + suchthat ε >, B ( B ϕ, ϕ < b [ α { β( log( }] as < for ( ( ε (.. α [ log] (.5 By the propertes of α Δ we ca easly obta, f, lsup (. Propertes of B α [ log( ].The set B s octat- le. O the other had fro the efto 3, we get for. the boudary pots of the set B (ϕ < ε <, for a certa f, hypersurface S S (ϕ whch dvdes the hyperoctat > ( ε for soe α [ log( ] + to Two parts, oe whch the equalty (.5 s true ad the sequece, fro the above we other α(log f, I whch t s false.thus we call t the hypersurface of obta, l sup geeralzed assocated typesof the plursubharoc fucto log( (.3 the class E (β, ad ay syste of ubers Cobg Eqs. (. ad (.3 we arrve at our result. (,,... S (ϕ Theore.: Let f (Z be a aalytc fucto havg wll be called a syste of geeralzed assocated types of the fucto. geeralzed order ad type the δ, where; ear: Cosderg α ( x log x, ad β ( x x ad ϕ( (, κ where δ lsup (.4 (, ax f ( Z, for every, the above defto κ z cocdes wth that of Jueja ad Kapoor[]. log α efto 6: For a fucto f (Z aalytc a ut polydsc exp log( c d ( κ we defe, [ q] l sup t log( t κ ( + +... + ad where ax ( r,, < t <. (. ad r d ( sup Z efto 7:For a fucto f, < < ; Z ( q Furtherore f, T lsup (.3. dα( x t ( t for l arg e xthe d log( x efto 8: Let α Δ ad β L, the the geeralzed δ, hece δ, provded the growth of the fucto order of ϕ β wth respect to the varable (eepg the others j fxed s defed α [ log ] s slower tha the growth of II.AIN ESULTS INTENATIONAL JOUNAL OF APPLIE ATHEATICS AN INFOATICS
Issue, Volue, 7 5 α [ log( ]. Proof: The proof follows fro the deftos of lt superor as well as careful use of Cauchy s Iequalty. ear: The above Theore holds true for aalytc fuctos havg postve order oly. Theore.: Let α Δ ad β L the s a Covex Fucto of β log(,... β log(. Proof: Choosg, t, s for,...(, ad t s ad puttg Eq.(. we get, ϕ[ exp{ β ( λβ log( + μβ (log( },..., ] λϕ (,,..., + μϕ (,,..., Therefore, ϕ exp lsup α [ β log( ] λϕ(,,..., + μϕ(,,..., (.5, [ { β ( λβ log( + μβ(log( },..., ] Provg the asserto. Theore.3: Let ϕ β ad B be the doa cosstg of the teror pots of the correspodg set B. The the age of the doa B. uder the ap ( b β (log( b s a covex doa provded b <. Proof: Explotg the equalty(. wth the followg values of t ad s as, β(log( + ( t exp β μ (.6, ( log( log( β b β b ( s exp β (.7. β (log( + λ ( β log( b β log( b But λβ log( t + μβ ( s β log( Therefore the result s obtaed upo substtutg the above values of t ad s (.. ear: The above Theore s sgfcatly dfferet fro what has bee obtaed []. Lea.: Let α Δ. the the ecessary codto for a pot (, b,... b to le the teror of the set b + B ( s lsup κ where κ f b Proof: Sce,. ϕ < log c < b log exp α( (.8. b α [ log( ] f < Cε expbα [ log( ] we ca get, for soe C >. (.9 Fro Cauchy s Iequalty, f C Ad choosg ε to be the root of the equato, exp α ( ( We get, C < Cε exp(( b ε exp α( b. (., Where fro the result follows upo tag lt superor. Theore.4:For the aalytc fucto f (Z log b l sup (. log + Proof: (f part We cosder the fucto ϕ ( Z, w of (+ coplex varables ad wrte t as Z w ϕ (, P ( Z w where P ( Z Z. The Cauchy s equalty alog wth the defto 6 results INTENATIONAL JOUNAL OF APPLIE ATHEATICS AN INFOATICS
Issue, Volue, 7 53 (, P Aexp [ q ] ( + + ε + ε + θε + ε + s exp β log( β λl + μ (.7 for,, (-. Whch upo ajorsg uder lt superor results log b l sup. (. log + (oly f part: For ths we frst assue, log b l sup μ <.Now usg the log defto of l sup we arrve at, t ( Ct ( C + < ε + +. [ q ] μ+ ε exp ( where C ad C are arbtrary cos ta ts. Now the fucto ( + F( Z [ q ] ( μ+ ε exp Z s aalytc. Now upo usg efto 6 μ we arrve at, whch ples, μ μ (.3. + Theore.4: For a fucto f (Z, aalytc ad satsfyg < < ad, [ q ] T l sup (.4 t ( t where q,3, s such that (q ad (q <, the, [ q ] b + log ( T l sup (.5. Proof: The proof follows the sae patter of the proof of Theore.5 ad hece s otted. Theore.5: Let α Δ ad s of the for α ( x λ(log( x, the for ay λ > ad β L, < γ γ >. Proof: Choosg l t exp β log( β λl + μ (.6 ad l t exp β log( β λl + μ (.8 s exp β log( β λl + μ (.9 l ε < ε <, (. l β log( C( for,,... ; (. C( + β log( ε (. λ C(, μ λ β log( (.3 ad usg the equalty (. wth the above values, ad upo solvg the olear equato α ( x λ(log( x for x we arrve at our result by usg the defto of lt superor. III. CONCLUSION (a The Growth of a Class of plursubharoc fuctos are extesvely used the Value strbuto Theory of fuctos of Several Coplex Varables. (b Plursubharoc fuctos are the hgher desoal geeralzato of sub haroc fuctos. (c I coplex aalyss, plursubharoc fuctos are used to descrbe pseudo covex doas, doas of holoorphy ad Ste afolds. IV. ACKNOWLEGEENTS The author expresses hs heartfelt thas to the eferees ad Prof. astores as well as the whole of WSEAS Staff,as well as to r. Al Kuar of KIST ICT. EFEENCES: [] Jueja,O.P. Ad Kapoor G.P. Aalytc fuctos- growth aspect, esearch Notes atheatcs, Pta 4,(985. [] Sha, Artasu O the growth of a class of Plursubharoc Fuctos a Ut dsc. J. Ida,Ist,Sc,69, Nov-ec.989,44-448. INTENATIONAL JOUNAL OF APPLIE ATHEATICS AN INFOATICS
Issue, Volue, 7 54 [3] Sha, Artasu Kuar Soe studes ad applcatos o certa growth aspects of etre fuctos of several coplex Varables Ph.. Thess IIT Kapur Aprl 984. [4] udger W.Brau et al: Local radal Phraga- Ldelof estates for plursubharoc fuctos o aalytc varetes. Proc. Aer. ath.soc. 3 (3,43-433. [5] A. Lexader et al. Idcators for plursubharoc fuctos of logarthc growth.uj.daa.edu 5 (. [6] Sereeta,.N. O the coecto betwee the growth of the axu odulus of a etre fucto Ad odul of the coeffcets of ts Power seres expaso. A. ath. Soc. Trasl.,97 88,9-3. [7] Lae A.P. ad Bolger WE. Edoscopc trasaxllary bopsy of pterygopalate space asses: A.prelary report. A J. hol 6:9-,. Edoscopc trasaxllary bopsy INTENATIONAL JOUNAL OF APPLIE ATHEATICS AN INFOATICS