International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 1 (2014), 68-80 http://www.etamaths.com GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC SANJIB KUMAR DATTA1,∗, TANMAY BISWAS2, SOUMEN KANTI DEB3 Abstract. In this paper we study some growth properties of composite func- tions analytic in the unit polydisc. Some results related to the generalised n variables based p-th Nevanlinna order (generalised n variables based p-th Nevanlinna lower order) and the generalised n variables based p-th Nevanlin- na relative order (generalised n variables based p-th Nevanlinna relative lower order) of an analytic function with respect to an entire function are established in this paper where n and p are any two positive integers. In fact in this paper we extend some results of [3] and [4]. 1. Introduction, Definitions and Notations. A function f analytic in the unit disc U = {z : |z| < 1} is said to be of finite Nevanlinna order [6] if there exists a number µ such that the Nevanlinna characteristic function Tf (r) = 1 2π ∫ 2π 0 log+ |f ( reiθ ) |dθ satisfies Tf (r) < (1 −r) −µ for all r in 0 < r0 (µ) < r < 1. The greatest lower bound of all such numbers µ is called the Nevanlinna order of f. Thus the Nevanlinna order ρf of f is given by ρf = lim sup r→1 log Tf (r) − log (1 −r) . Similarly, the Nevanlinna lower order λf of f are given by λf = lim inf r→1 log Tf (r) − log (1 −r) . L. Bernal introduced the relative order between two entire functions of single variables to avoid comparing growth just with the exponential function exp z. In this connection, Banerjee and Dutta [2] gave the following definition in a unit disc: Definition 1. [2] If f be analytic in U and g be entire , then the relative order of f with respect to g denoted by ρg (f) is defined by ρg (f) = inf { µ > 0 : Tf (r) < Tg [( 1 1 −r )µ] for all 0 < r0 (µ) < r < 1 } . 2010 Mathematics Subject Classification. 30D20, 30D30, 32A15. Key words and phrases. Growth; analytic function of n complex variables; composite function; generalised n variables based p-th Nevanlinna order; generalised n variables based p-th Nevanlinna relative order; Unit Polydisc. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 68 GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 69 Similarly, one may define λg (f), the relative lower order of f with respect to g. With g (z) = exp z, the definition coincides with the definition of Nevanlinna order of f. Analogously, λg (f) = lim inf r→1 log T−1g Tf (r) − log (1 −r) . Extending the notion of single variables to several variables, let f(z1,z2, · · ·,zn) be a non-constant analytic function of n complex variables z1,z2, ··· zn−1 and zn in the unit polydisc U = {(z1,z2, · · ·,zn) : |zj| ≤ 1, j = 1, 2, · · ·,n; r1 > 0,r2 > 0, · · ·rn > 0} . Now in the line of Nevanlinna order [6], in this paper we introduce the generalised n variables based p-th Nevanlinna order and the generalised n variables p-th Nevanlinna lower order for functions of n complex variables analytic in a unit polydisc as follows : ρ [p] f (r1,r2, ...,rn) = lim sup r1,r2,...rn→1 log[p] Tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] and λ [p] f (r1,r2, ...,rn) = lim inf r1,r2,...rn→1 log[p] Tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] where log[k] x = log(log[k−1] x) for k = 1, 2, 3, ... and log[0] x = x. When n = p = 1, the above definition reduces to the definition of Juneja and Kapoor [6]. Likewise, one may introduce the generalised n variables based p-th relative Nevanlinna order ( generalised n variables based p-th relative Nevanlinna lower or- der) for functions of n complex variables analytic in a unit polydisc in the following manner : Definition 2. Let Tf (r1,r2, ...,rn) denote the Nevanlinna’s characteristic function of f of n variables. The generalised n variables based p-th relative Nevanlinna order ρ [p]f g (r1,r2, ...,rn) and generalised n variables based p-th relative Nevanlinna lower order λ[p]fg (r1,r2, ...,rn) of an analytic function f in U with respect to another entire function g in n complex variables are defined in the following way : ρ[p]fg (r1,r2, ...,rn) = lim sup r1,r2→∞ log[p] T−1g Tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] and λ[p]fg (r1,r2, ...,rn) = lim inf r1,r2→∞ log[p] T−1g Tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] where n and p are any two positive integers . If we consider p = n = 1 in Definition 2, then it coincides with Definition 1. In the paper we establish some results relating to the composition of two non- constant analytic functions, of n complex variables in the unit polydisc U = {(z1,z2, · · ·,zn) : |zj| ≤ 1, j = 1, 2, · · ·,n; r1 > 0,r2 > 0, · · ·rn > 0} . Also we prove a few theorems related to generalised n variables based p-th rela- tive Nevanlinna order ρ [p]f g (r1,r2, ...,rn) (generalised n variables based p-th relative 70 DATTA, BISWAS AND DEB Nevanlinna lower order λ[p]fg (r1,r2, ...,rn) ) of an analytic function f with respect to an entire function g of n complex variables which are in fact some extensions of earlier results as proved in [3] and [4]. We do not explain the standard definitions and notations in the theory of entire functions of severable variables as those are available in [1], [5] and [7]. 2. Theorems. In this section we present the main results of the paper. Theorem 1. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U such that 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[q] g (r1,r2, ...,rn) ≤ ρ[q]g (r1,r2, ...,rn) < ∞. Then λ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) where p and q are any two positive integers . Proof. From the definition of generalised n variables based p-th Nevanlinna order and generalised n variables based p-th Nevanlinna lower order of analytic functions in the unit polydisc U, we have for arbitrary positive � and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] Tf◦g (r1,r2, ...,rn)(1) ≥ ( λ[p] f◦g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] and log[q] Tg (r1,r2, ...,rn)(2) ≤ ( ρ[q] g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . Now from (1) and (2) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) − � ρ[q] g (r1,r2, ...,rn) + � . As � (> 0) is arbitrary, we obtain that (3) lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . Again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] Tf◦g (r1,r2, ...,rn)(4) ≤ ( λ[p] f◦g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 71 and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[q] Tg (r1,r2, ...,rn)(5) ≥ ( λ[q] g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . So combining (4) and (5) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) + � λ[q] g (r1,r2, ...,rn) − � . Since � (> 0) is arbitrary, it follows that (6) lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . Also for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, we get that log[q] Tg (r1,r2, ...,rn)(7) ≤ ( λ[q] g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . Now from (1) and (7) , we obtain for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) − � λ[q] g (r1,r2, ...,rn) + � . Choosing � → 0, we get that (8) lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . Also for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] Tf◦g (r1,r2, ...,rn)(9) ≤ ( ρ[p] f◦g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . So from (5) and (9) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) + � λ[q] g (r1,r2, ...,rn) − � . As � (> 0) is arbitrary, we obtain that (10) lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . 72 DATTA, BISWAS AND DEB Thus the theorem follows from (3), (6), (8) and (10). The following theorem can be proved in the line of Theorem 1 and so its proof is omitted. Theorem 2. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U with 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[l] f (r1,r2, ...,rn) ≤ ρ[l]f (r1,r2, ...,rn) < ∞ where p and l are any two positive integers. Then λ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) . Theorem 3. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U such that 0 < ρ[p] f◦g (r1,r2, ...,rn) < ∞ and 0 < ρ[q] g (r1,r2, ...,rn) < ∞. Then lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) where p and q are any two positive integers . Proof. From the definition of generalised n variables based p-th Nevanlinna order, we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[q] Tg (r1,r2, ...,rn)(11) ≥ ( ρ[q] g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . Now from (9) and (11) , it follows for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) + � ρ[q] g (r1,r2, ...,rn) − � . As � (> 0) is arbitrary, we obtain that (12) lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . Again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] Tf◦g (r1,r2, ...,rn)(13) ≥ ( ρ[p] f◦g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 73 So combining (2) and (13) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ ρ[p] f◦g (r1,r2, ...,rn) − � ρ[q] g (r1,r2, ...,rn) + � . Since � (> 0) is arbitrary, it follows that (14) lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≥ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . Thus the theorem follows from (12) and (14) . The following theorem can be carried out in the line of Theorem 3 and therefore we omit its proof. Theorem 4. Let f and g be any two non-constant analytic functions of n com- plex variables in the unit polydisc U with 0 < ρ[p] f◦g (r1,r2, ...,rn) < ∞ and 0 < ρ[l] f (r1,r2, ...,rn) < ∞ where p and l are any two positive integers. Then lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) . The following theorem is a natural consequence of Theorem 1 and Theorem 3: Theorem 5. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U such that 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[q] g (r1,r2, ...,rn) ≤ ρ[q]g (r1,r2, ...,rn) < ∞. Then lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) ≤ min { λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) } ≤ max { λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) where p and q are any two positive integers . Analogously one may state the following theorem without its proof. Theorem 6. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U with 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[l] f (r1,r2, ...,rn) ≤ ρ[l]f (r1,r2, ...,rn) < ∞ where p and l are any two 74 DATTA, BISWAS AND DEB positive integers .Then lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) ≤ min { λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) } ≤ max { λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) . Theorem 7. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U such that ρ[l] f (r1,r2, ...,rn) < ∞ and λ[p]f◦g (r1,r2, ...,rn) = ∞. Then lim r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[l] Tf (r1,r2, ...,rn) = ∞ where p and l are any two positive integers . Proof. Let us suppose that the conclusion of the theorem do not hold. Then we can find a constant β > 0 such that for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, (15) log[p] Tf◦g (r1,r2, ...,rn) ≤ β log[l] Tf (r1,r2, ...,rn) . Again from the definition of ρ[l] f (r1,r2, ...,rn) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[l] Tf (r1,r2, ...,rn)(16) ≤ [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . Thus from (15) and (16) , we have for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] Tf◦g (r1,r2, ...,rn) ≤ β [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] i.e., log[p] Tf◦g (r1,r2, ...,rn) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] ≤ β [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] i.e., lim inf r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] = λ[p] f◦g (r1,r2, ...,rn) < ∞. This is a contradiction. Hence the theorem follows. GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 75 Remark 1. Theorem 7 is also valid with “limit superior” instead of “limit” if λ[p] f◦g (r1,r2, ...,rn) = ∞ is replaced by ρ[p]f◦g (r1,r2, ...,rn) = ∞ and the other condi- tions remain the same. Corollary 8. Under the assumptions of Theorem 7 and Remark 1, lim r1,r2,...rn→1 Tf◦g (r1,r2, ...,rn) Tf (r1,r2, ...,rn) = ∞ and lim sup r1,r2,...rn→1 Tf◦g (r1,r2, ...,rn) Tf (r1,r2, ...,rn) = ∞ respectively hold if p = l. The proof is omitted. Analogously one may also state the following theorem and corollaries without their proofs as those may be carried out in the line of Remark 1, Theorem 7 and Corollary 8 respectively. Theorem 9. Let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc U with ρ[q] g (r1,r2, ...,rn) < ∞ and ρ[p]f◦g (r1,r2, ...,rn) = ∞ where p and q are any two positive integers. Then lim sup r1,r2,...rn→1 log[p] Tf◦g (r1,r2, ...,rn) log[q] Tg (r1,r2, ...,rn) = ∞ . Corollary 10. Theorem 9 is also valid with “limit” instead of “limit superior” if ρ[p] f◦g (r1,r2, ...,rn) = ∞ is replaced by λ[p]f◦g (r1,r2, ...,rn) = ∞ and the other conditions remain the same. Corollary 11. Under the assumptions of Theorem 7 and Corollary 10, lim sup r1,r2,...rn→1 Tf◦g (r1,r2, ...,rn) Tg (r1,r2, ...,rn) = ∞ and lim r1,r2,...rn→1 Tf◦g (r1,r2, ...,rn) Tg (r1,r2, ...,rn) = ∞ respectively hold if p = q. In the next three theorems we establish some comparative growth properties related to the generalised n variables based p-th relative Nevanlinna order (gener- alised n variables based p-th relative Nevanlinna lower order) of an analytic function with respect to an entire function in the unit poly disc U. Theorem 12. Let f,h be any two analytic functions of n complex variables in U and g be entire in n complex variables such that 0 < λ[p]fg (r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < λ[p]hg (r1,r2, ...,rn) ≤ ρ [p]h g (r1,r2, ...,rn) < ∞. Then λ[p]fg (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) where p is any positive integer. Proof. From the definition of generalised n variables based p-th relative Nevanlinna order and generalised n variables based p-th relative Nevanlinna lower order of an analytic function with respect to an entire function in an unit polydisc U, we have 76 DATTA, BISWAS AND DEB for arbitrary positive � and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] T−1g Tf (r1,r2, ...,rn)(17) ≥ [ λ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] and log[p] T−1g Th(r1,r2, ...,rn)(18) ≤ [ ρ[p]hg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . Now from (17) and (18) , it follows for all sufficiently large values of ( 1 1−r1 ) ,( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) − � ρ [p]h g (r1,r2, ...,rn) + � . As � (> 0) is arbitrary, we obtain that (19) lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . Again we have for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that (20) log[p] T−1g Tf (r1,r2, ...,rn) ≤ [ λ[p]fg (r1,r2, ...,rn) + � ] [− log (1 −r)] and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] T−1g Th(r1,r2, ...,rn)(21) ≥ [ λ[p]hg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . So combining (20) and (21) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , tending to infinity that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) + � λ[p]hg (r1,r2, ...,rn) − � . Since � (> 0) is arbitrary, it follows that (22) lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . Also for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] T−1g Th(r1,r2, ...,rn)(23) ≤ [ λ[p]hg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 77 Now from (17) and (23) , we obtain for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , tending to infinity that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) − � λ[p]hg (r1,r2, ...,rn) + � . Choosing � (> 0) , we get that (24) lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . Also for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] T−1g Tf (r1,r2, ...,rn)(25) ≤ [ ρ[p]fg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . So from (21) and (25) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) + � λ[p]hg (r1,r2, ...,rn) − � . As � (> 0) is arbitrary, we obtain from above that (26) lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . Thus the theorem follows from (19) , (22) , (24) and (26) . Theorem 13. Let f,h be any two analytic functions of n complex variables in U and g be entire in n complex variables with 0 < ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < ρ [p]h g (r1,r2, ...,rn) < ∞ where p is any positive integer. Then lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) . Proof. From the definition of generalised n variables based p-th relative Nevanlinna order, we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] T−1g Th(r1,r2, ...,rn)(27) ≥ [ ρ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . 78 DATTA, BISWAS AND DEB Now from (25) and (27) , it follows for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) + � ρ [p]h g (r1,r2, ...,rn) − � . As � (> 0) is arbitrary, we obtain that (28) lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . Again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] T−1g Tf (r1,r2, ...,rn)(29) ≥ [ ρ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . So combining (18) and (29) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ ρ [p]f g (r1,r2, ...,rn) − � ρ [p]h g (r1,r2, ...,rn) + � . Since � (> 0) is arbitrary, it follows that (30) lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≥ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . Thus the theorem follows from (28) and (30) . In view of Theorem 12 and Theorem 13, we may state the following theorem without its proof. Theorem 14. Let f,h be any two analytic functions of n complex variables in U and g be entire in n complex variables such that 0 < λ[p]fg (r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < λ[p]hg (r1,r2, ...,rn) ≤ ρ [p]h g (r1,r2, ...,rn) < ∞. Then lim inf r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) ≤ min { λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) , ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) } ≤ max { λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) , ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] T−1g Tf (r1,r2, ...,rn) log[p] T−1g Th(r1,r2, ...,rn) where p is any positive integer. GROWTH ANALYSIS OF FUNCTIONS ANALYTIC IN THE UNIT POLYDISC 79 Theorem 15. Let f,h be any two analytic functions of n complex variables in U and g be entire in n complex variables such that ρ [p]f h (r1,r2, ...,rn) < ∞ and λ [p]f◦g h (r1,r2, ...,rn) = ∞ where p is any positive integer. Then lim r1,r2,...rn→1 log[p] T−1h Tf◦g(r1,r2, ...,rn) log[p] T−1h Tf (r1,r2, ...,rn) = ∞ . The proof is omitted because it can be carried out using the same technique as involved in Theorem 7. Remark 2. Theorem 15 is also valid with “limit superior” instead of “limit” if λ [p]f◦g h (r1,r2, ...,rn) = ∞ is replaced by ρ [p]f◦g h (r1,r2, ...,rn) = ∞ and the other conditions remain the same. Corollary 16. Under the assumptions of Theorem 15 and Remark 2, lim r1,r2,...rn→1 T−1h Tf◦g(r1,r2, ...,rn) T−1h Tf (r1,r2, ...,rn) = ∞ and lim sup r1,r2,...rn→1 T−1h Tf◦g(r1,r2, ...,rn) T−1h Tf (r1,r2, ...,rn) = ∞ respectively hold. The proof is omitted. Similarly, one may also state the following theorem and corollaries without their proofs as they may be carried out in the line of Remark 2, Theorem 15 and Corollary 16 respectively. Theorem 17. Let f,h be any two analytic functions of n complex variables in U and g be entire in n complex variables such that ρ [p]g h (r1,r2, ...,rn) < ∞ and ρ [p]f◦g h (r1,r2, ...,rn) = ∞. Then lim sup r1,r2,...rn→1 log[p] T−1h Tf◦g(r1,r2, ...,rn) log[p] T−1h Tg(r1,r2, ...,rn) = ∞ where p is any positive integer. Corollary 18. Theorem 17 is also valid with “limit” instead of “limit superior” if ρ [p]f◦g h (r1,r2, ...,rn) = ∞ is replaced by λ [p]f◦g h (r1,r2, ...,rn) = ∞ and the other conditions remain the same. Corollary 19. Under the assumptions of Theorem 15 and Corollary 18, lim sup r1,r2,...rn→1 T−1h Tf◦g(r1,r2, ...,rn) T−1h Tg(r1,r2, ...,rn) = ∞ and lim r1,r2,...rn→1 T−1h Tf◦g(r1,r2, ...,rn) T−1h Tg(r1,r2, ...,rn) = ∞ respectively hold. References [1] Agarwal, A. K., On the properties of an entire function of two complex variables, Canadian J.Math. Vol. 20 (1968), pp.51–57. [2] Banerjee, D. and Dutta, R. K., Relative order of functions analytic in the unit disc, Bull. Cal. Math. Soc. Vol. 101, No. 1 (2009), pp. 95 - 104. [3] Datta, S. K. and Deb, S. K. , Growth properties of functions analytic in the unit disc, Inter- national J. of Math. Sci & Engg. Appls (IJMSEA), Vol. 3, No. IV (2009), pp. 2171-279. [4] Datta, S. K. and Jerine, E., On the generalised growth properties of functions analytic in the unit disc, Wesleyan Journal of Research, Vol.3, No.1 (2010), pp.13-19. 80 DATTA, BISWAS AND DEB [5] Fuks, B. A., Theory of analytic functions of several complex variables, Moscow, 1963. [6] Juneja, O. P. and Kapoor, G.P., Analytic functions-growth aspects, Pitman advanced pub- lishing program, 1985. [7] Kiselman, C. O., Plurisubharmonic functions and potential theory in several complex variables, a contribution to the book project, Development of Mathematics 1950-2000, edited by Jean Paul Pier. 1Department of Mathematics,University of Kalyani, Kalyani, Dist-Nadia,PIN- 741235, West Bengal, India 2Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.- Krishnagar, Dist-Nadia, PIN- 741101, West Bengal, India 3Bahin High School, P.O.-Bahin,Dist.-Uttar Dinajpur, PIN-733157, West Bengal, In- dia ∗Corresponding author