GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX VARIABLES

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1 Sa Communications in Matematical Analysis (SCMA Vol. 3 No. 2 ( ttp://scma.marae.ac.ir GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX VARIABLES SANJIB KUMAR DATTA 1 AND TANMAY BISWAS 2 Abstract. In tis paper we introduce te idea o eneralized relative order (respectively eneralized relative lower order o entire unctions o two complex variables. Hence we study some rowt properties o entire unctions o two complex variables on te basis o te deinition o eneralized relative order eneralized relative lower order o entire unctions o two complex variables. 1. Introduction Deinitions Notations Let be an entire unction o two complex variables wic is olomorpic in te closed polydisc U = {(z 1 z 2 : z i r i i = 1 2 or all r 1 0 r 2 0} M (r 1 r 2 = max { (z 1 z 2 : z i r i i = 1 2}. Ten in view o maximum principal Hartos teorem ([5] p. 2 p. 51 M (r 1 r 2 is an increasin unction o r 1 r 2. In te sequel te ollowin two notations are used: lo [k] x = lo lo [k 1] x or k = ; lo [0] x = x exp [k] x = exp exp [k 1] x or k = ; exp [0] x = x. Te ollowin deinition is well known: 2010 Matematics Subject Classiication. 32A15 30D20. Key words prases. Entire unctions Generalized relative order Generalized relative lower order Two complex variables Composition Growt. Received: 15 November 2015 Accepted: 09 January Correspondin autor. 13

2 14 S. K. DATTA AND T. BISWAS Deinition 1.1 ([5] p. 339 (see also [1]. Te order v2 ρ te lower order v2 λ o an entire unction o two complex variables are deined as ρ = lim sup r 1 r 2 lo [2] M (r 1 r 2 lo [2] M (r 1 r 2 λ = lim in. r 1 r 2 I we consider te above deiniton or te case o sinle variable ten te deinition coincides wit te classical deinition o order (see [12] wic is as ollows: Deinition 1.2 ([12]. Te order ρ te lower order λ o an entire unction are deined in te ollowin ways: ρ = lim sup r lo [2] M (r lo r lo [2] M (r λ = lim in r lo r were M (r = max { (z : z = r}. I is non-constant ten M (r is strictly increasin continuous its inverse M 1 : ( (0 (0 exists is suc tat lim M 1 (s =. Bernal ([2] [3] introduced te deinition o relative s order o wit respect to denoted by ρ ( as ollows: ρ ( = in {µ > 0 : M (r < M (r µ or all r > r 0 (µ > 0} = lim sup r lo M 1 M (r. lo r Te deinition coincides wit te classical one [12] i (z = exp z. Durin te past decades several autors (see [ ] made close investiations on te properties o relative order o entire unctions o sinle variable. In te case o relative order it was ten natural or Banerjee Dutta [4] to deine te relative order o entire unctions o two complex variables as ollows: Deinition 1.3 ([4]. Te relative order between two entire unctions o two complex variables denoted by v2 ρ ( is deined as: ρ ( = in {µ > 0 : M (r 1 r 2 < M (r µ 1 rµ 2 ; r 1 R (µ r 2 R (µ} lo M 1 M (r 1 r 2 = lim sup r 1 r 2

3 GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX were are entire unctions olomorpic in te closed polydisc U = {(z 1 z 2 : z i r i i = 1 2 or all r 1 0 r 2 0} te deinition coincides wit Deinition 1.1 (see [4] i (z 1 z 2 = exp (z 1 z 2. However an entire unction o two complex variables or wic order lower order are te same is said to be o reular rowt. Te unction exp (z 1 z 2 is an example o reular rowt o entire unctions o two complex variables. Furter te unctions wic are not o reular rowt are said to be o irreular rowt. Now in te line o Juneja Kapoor Bajpai [7] we would like to introduce te deinitions o (p q-t order (p q-t lower order o an entire unction o two complex variables respectively as ollows: lo [p] M (r 1 r 2 ρ (p q = lim sup r 1 r 2 lo [q] (r 1 r 2 lo [p] M (r 1 r 2 λ (p q = lim in r 1 r 2 lo [q] (r 1 r 2 were p q are any two positive inteers wit p q. In particular i we consider q = 1 ten te above deinition is reduced to te ollowin deinitions o eneralized order eneralized lower order in connection wit two complex variables: Deinition 1.4. Te eneralized order v2 te eneralized lower order v2 o an entire unction o two complex variables are deined as were p 1. lo [p] M (r 1 r 2 = lim sup r 1 r 2 = lim in r 1 r 2 lo [p] M (r 1 r 2 Tese deinitions extend te eneralized order eneralized lower order o an entire unction as considered in [11]. Furter an entire unction o two complex variables is said to be o reular (p q-rowt i its (p q-t order coincides wit its (p q-t lower order oterwise is said to be o irreular (p q-rowt.

4 16 S. K. DATTA AND T. BISWAS Now in te case o relative order (respectively relative lower order it is ten natural to deine te eneralized relative order (respectively eneralized relative lower order o entire unctions o two complex variables as ollows: Deinition 1.5. Let (z 1 z 2 (z 1 z 2 be any two entire unctions o two complex variables z 1 z 2 wit maximum modulus unctions M (r 1 r 2 M (r 1 r 2 respectively ten or any positive interer p te eneralized relative order (respectively eneralized relative lower order o wit respect to denoted by v2 is deined as ( 1 M (r 1 r 2 ( = lim sup r 1 r 2 respectively v2 ( = lim in r 1 r 2 ( (respectively v2 ( lo M 1 M (r 1 r 2 In tis paper we wis to prove some results related to te rowt properties o composite entire unctions o two complex variables on te basis o eneralized relative order eneralized relative lower order o entire unctions o two complex variables. We do not explain te stard deinitions notations in te teory o entire unction o two complex variables as tose are available in [5]. 2. Teorems In tis section we present te main results o te paper. Teorem 2.1. Let be any two entire unctions o two complex variables wit 0 < v2 v2 < 0 < v2 λ (m p ρ (m p < were m p are any positive inteers wit m p. Ten ρ (m p ( [m] min λ (m p ρ ρ (m p [m] max λ (m p ρ ρ (m p v2 ρ [p] (.

5 GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX ρ[m] λ (m p. v2 we ave or all sui- Proo. From te deinitions o v2 ciently lare values o r 1 r 2 tat (2.1 (2.2 M (r 1 r 2 exp [m] {( M (r 1 r 2 exp [m] {( } + ε } ε also or a sequence o values {( o r 1 r 2 tendin to ininity we et tat } (2.3 M (r 1 r 2 exp [m] ε { } (2.4 M (r 1 r 2 exp [m] + ε. Similarly rom te deinitions o v2 ρ (m p v2 λ (m p it ollows or all suiciently lare values { o r 1 r 2... r n tat } M (r 1 r 2 exp [m] ( v2 ρ (m p + ε lo [p] (r 1 r 2 [ { }] (r 1 r 2 M 1 exp [m] ( v2 ρ (m p + ε lo [p] (r 1 r 2 [ ] (2.5 M 1 (r 1 r 2 exp [p] lo [m] (r 1 r 2 ( v2 ρ (m p + ε { } M (r 1 r 2 exp [m] ( v2 λ (m p ε lo [p] (r 1 r 2 [ ] (2.6 M 1 (r 1 r 2 exp [p] lo [m] (r 1 r 2 ( v2 λ (m p ε or a sequence o values o { r 1 r 2 tendin to ininity we obtain tat } M (r 1 r 2 exp [m] ( n2 ρ (m p ε lo [p] (r 1 r 2 [ ] (2.7 M 1 (r 1 r 2 exp [p] lo [m] (r 1 r 2 ( n2 ρ (m p ε { } M (r 1 r 2 exp [m] ( n2 λ (m p + ε lo [p] (r 1 r 2

6 18 S. K. DATTA AND T. BISWAS [ ] (2.8 M 1 (r 1 r 2 exp [p] lo [m] (r 1 r 2. ( n2 λ (m p + ε Now rom (2.3 in view o (2.5 we et or a sequence o values o r 1 r 2 tendin to ininity we et tat [ { }] 1 M (r 1 r 2 1 exp [m] ε {( M (r 1 r 2 lo [p] exp [p] lo[m] exp [m] ( v2 ρ (m p + ε 1 M (r 1 r 2 1 M (r 1 r 2 As ε (> 0 is arbitrary it ollows tat (2.9 ( ρ[m] ρ (m p. ε ε ( v2 ρ (m p + ε lo (r 1r 2 ε ( v2 ρ (m p + ε. } Analoously rom (2.2 in view o (2.8 or a sequence o values o r 1 r 2 tendin to ininity we et tat [ { }] 1 M (r 1 r 2 1 exp [m] ε {( M (r 1 r 2 lo [p] exp [p] lo[m] exp [m] ( v2 λ (m p + ε ε } 1 M (r 1 r 2 1 M (r 1 r 2 ε ( v2 λ (m p + ε lo (r 1r 2 ε ( v2 λ (m p + ε.

7 GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX Since ε (> 0 is arbitrary we et rom above tat (2.10 ( λ[m] λ (m p. Aain in view o (2.6 we ave rom (2.1 or all suiciently lare values o r 1 r 2 tat [ { }] 1 M (r 1 r 2 1 exp [m] + ε {( M (r 1 r 2 lo [p] exp [p] lo[m] exp [m] ( v2 λ (m p ε 1 M (r 1 r 2 + ε + ε ( v2 λ (m p ε lo (r 1r 2 1 M (r 1 r 2 + ε ( v2 λ (m p ε. Since ε (> 0 is arbitrary we obtain tat (2.11 ( ρ[m] λ (m p. } Aain rom (2.2 in view o (2.5 wit te same reasonin we et tat (2.12 ( λ[m] ρ (m p. Also in view o (2.7 we et rom (2.1 or a sequence o values o r 1 r 2 tendin to ininity tat [ { }] 1 M (r 1 r 2 1 exp [m] + ε {( M (r 1 r 2 lo [p] exp [p] lo[m] exp [m] ( v2 ρ (m p ε + ε } 1 M (r 1 r 2 + ε ( v2 ρ (m p ε lo (r 1r 2

8 20 S. K. DATTA AND T. BISWAS 1 M (r 1 r 2 + ε ( v2 ρ (m p ε. Since ε (> 0 is arbitrary we et rom above tat (2.13 ( ρ[m] ρ (m p. Similarly rom (2.4 in view o (2.6 it ollows or a sequence o values o r 1 r 2 tendin to ininity we et tat [ { }] 1 M (r 1 r 2 1 exp [m] + ε {( M (r 1 r 2 lo [p] exp [p] lo[m] exp [m] ( v2 λ (m p ε 1 M (r 1 r 2 1 M (r 1 r 2 + ε + ε ( v2 λ (m p ε lo (r 1r 2 + ε ( v2 λ (m p ε. As ε (> 0 is arbitrary we obtain rom above tat (2.14 λ [p] ( λ[m] λ (m p. } Te teorem ollows rom (2.9 (2.10 (2.11 (2.12 (2.13 (2.14. Corollary 2.2. Let be an entire unction o two complex variables wit eneralized order v2 eneralized lower order v2 were m is any positive inteer. Also let be an entire unction o two complex variables wit reular (m p-rowt were p m are all positive inteers suc tat m p. Ten ( = λ[m] ρ (m p ( = ρ[m] ρ (m p.

9 GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX Corollary 2.3. Let be any two entire unctions o two complex variables wit reular eneralized rowt reular (m p rowt respectively were p m are all positive inteers wit m p. Ten ( = v2 ( = ρ[m] ρ (m p. Corollary 2.4. Let be any two entire unctions o two complex variables wit reular eneralized rowt reular (m p rowt respectively were p m are all positive inteers wit m p. Also suppose tat v2 = v2 ρ (m p. Ten ( = v2 ( = 1. Corollary 2.5. Let be an entire unction o two complex variables wit eneralized order v2 eneralized lower order v2 were m is any positive inteer. Ten or any entire unction o two complex variables (i (ii (iii (iv ( = wen v2 ρ (m p = 0 ( = wen v2 λ (m p = 0 ( = 0 wen v2 ρ (m p = ( = 0 wen v2 λ (m p = were p is any positive inteer wit m p. Corollary 2.6. Let be an entire unction o two complex variables wit (m p-t order vn ρ (m p (m p-t lower order vn λ (m p were m p are positive inteers wit m p. Ten or any entire unction o two complex variables (i (ii (iii (iv ( = 0 wen v2 = 0 λ [p] ( = 0 wen v2 = 0 ( = wen v2 = ( = wen v2 =. Teorem 2.7. Let be any tree entire unctions o two complex variables suc tat v2 ( < ( = were p is any positive inteer. Ten lim M (r 1 r 2 r 1 r 2 M =. (r 1 r 2

10 22 S. K. DATTA AND T. BISWAS Proo. Let us suppose tat te conclusion o te teorem do not old. Ten we can ind a constant β > 0 suc tat or a sequence o values o r 1 r 2 tendin to ininity (2.15 M (r 1 r 2 β M (r 1 r 2. Aain rom te deinition o v2 ( it ollows or all suiciently lare values o r 1 r 2 tat ( (2.16 M (r 1 r 2 ( + ϵ. Tus rom (2.15 (2.16 we ave or a sequence o values o r 1 r 2 tendin to ininity tat ( M (r 1 r 2 β ( + ϵ ( M (r 1 r 2 β ( + ϵ lim in r 1 r 2 Tis is a contradiction. Tus te teorem ollows. M (r 1 r 2 = v2 ( <. Remark 2.8. Teorem 2.7 is also valid wit limit superior instead o limit i v2 ( = is replaced by ( = te oter conditions remain te same. Corollary 2.9. Under te assumptions o Teorem 2.7 Remark 2.8 lo [p 1] M 1 lim M (r 1 r 2 r 1 r 2 lo [p 1] M 1 M = (r 1 r 2 lo [p 1] M 1 lim sup M (r 1 r 2 r 1 r 2 lo [p 1] M 1 M (r 1 r 2 respectively old. Proo. Te proo is omitted. = Analoously one may also state te ollowin teorem remark corollary witout teir proos as tose may be carried out in te line o Remark 2.8 Teorem 2.7 Corollary 2.9 respectively.

11 GROWTH ANALYSIS OF ENTIRE FUNCTIONS OF TWO COMPLEX Teorem Let be any tree entire unctions o two complex variables wit v2 ( < ( = were p is any inteer. Ten lim sup M (r 1 r 2 r 1 r 2 M =. (r 1 r 2 Remark Teorem 2.10 is also valid wit limit instead o limit superior i v2 ( = is replaced by ( = te oter conditions remain te same. Corollary Under te assumptions o Teorem 2.10 Remark 2.11 lo [p 1] M 1 lim sup M (r 1 r 2 r 1 r 2 lo [p 1] M 1 M = (r 1 r 2 lo [p 1] M 1 lim M (r 1 r 2 r 1 r 2 lo [p 1] M 1 M = (r 1 r 2 respectively old. Reerences 1. A.K. Aarwal On te properties o entire unction o two complex variables Canadian Journal o Matematics 20 ( L. Bernal Crecimiento relativo de unciones enteras. Contribución al estudio de lasunciones enteras con índice exponencial inito Doctoral Dissertation University o Seville Spain L. Bernal Orden relativo de crecimiento de unciones enteras Collect. Mat. 39 ( D. Banerjee R. K. Dutta Relative order o entire unctions o two complex variables International J. o Mat. Sci. & En. Appls. (IJMSEA 1(1 ( A.B. Fuks Teory o analytic unctions o several complex variables Moscow S. Halvarsson Growt properties o entire unctions dependin on a parameter Annales Polonici Matematici 14(1 ( O.P. Juneja G.P. Kapoor S.K. Bajpai On te (pq-order lower (pq- order o an entire unction J. Reine Anew. Mat. 282 ( C.O. Kiselman Order type as measure o rowt or convex or entire unctions Proc. Lond. Mat. Soc. 66(3 ( C.O. Kiselman Plurisubarmonic unctions potential teory in several complex variable a contribution to te book project Development o Matematics edited by Hean-Paul Pier. 10. B.K. Lairi D. Banerjee A note on relative order o entire unctions Bull. Cal. Mat. Soc. 97(3 ( D. Sato On te rate o rowt o entire unctions o ast rowt Bull. Amer. Mat. Soc. 69 (

12 24 S. K. DATTA AND T. BISWAS 12. E.C. Titcmars Te teory o unctions 2nd ed. Oxord University Press Oxord Department o Matematics University o Kalyani P.O.-Kalyani Dist-Nadia PIN West Benal India. address: sanjib kr datta@yaoo.co.in 2 Rajbari Rabindrapalli R. N. Taore Road P.O.-Krisnaar Dist- Nadia PIN West Benal India. address: tanmaybiswas mat@redimail.com