Study of Growth Properties on the the Basis of Gol dberg Order of Composite Entire Functions of Several Variables

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1 Int. Journal of Math. Analysis, Vol. 5, 2011, no. 23, Study of Growth Properties on the the Basis of Gol dberg Order of Composite Entire Functions of Several Variables Sanjib Kumar atta epartment of Mathematics, University of Kalyani P.O.-Kalyani, ist-nadia, Pin West Bengal, India Former Address: (epartment of Mathematics, University of North Bengal P.O.-North Bengal University, Raja Rammohunpur ist-arjeeling, Pin West Bengal, India) sanjib kr s kr datta sk datta Meghlal Mallik Panighta U..M. High School P.O.-Paglachandi, ist Nadia, Pin West Bengal, India megh lal meghlal Abstract In the paper we discuss about the comparative growth properties related to the Gol dberg order of composition of two entire function of several complex variables. Mathematics Subject Classification: 3030, 3035 Keywords: Gol dberg order, composite entire function, several variable

2 1140 S. K. atta and M. Mallik 1 Introduction, efinitions and Notations. We denote complex and real n-space by C n and R n respectively and indicate the point (z 1,...z n ), (m 1,...m n )ofc n or I n by their corresponding unsuffixed symbols z, m respectively where I denotes the set of non-negative integers. The modulus of z, denoted by z, is defined as z =( z z n 2 ) 1 2. If the coordinates of the vector m are non-negative integers, then z m will denote z m zmn n and m = m m n. Let C n be an arbitrary bounded complete n-circular domain with center at the origin of coordinates. Then for the anlytic function f and R> 0, M f, (R) = sup f(z), where a point zɛ R if and only if z ɛ. R zɛ R efinition 1 {[2], p. 339} The Gol dberg order (briefly G-order) ρ f respect to the domain is defined as of f with ρ (f) ρ f = log [2] M f, (R) where log [k] x = log(log [k 1] x) for k =1, 2, 3,...and log [0] x = x. The lower Gol dberg order λ f of f with respect to the domain is defined as λ (f) λ f = log [2] M f, (R) We say that f is of regular growth if ρ f = λ f. efinition 2 [1] Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at the origin in C n. Then the relative G-order ρ g, (f) of f with respect to g and the domain is defined by ρ g, (f) = inf{μ >0; M f, (R) <M g, (R μ ), for all R>R 0 (μ) > 0}. If f is a non-constant entire function, then M f, (R) is a strictly increasing and continuous function of R and its inverse function M 1 f, :( f(0), ) (0, ) exists. It then easily follows that ρ g, (f) = log M 1 g, (M f, (R)).

3 Study of growth properties 1141 Similarly, the relative lower order λ g, (f) of f with respect to g and the domain is defined by λ g, (f) = log M 1 g, (M f, (R)). Throughout this paper we shall measure the growth of entire functions relative to the entire function g and will represent a bounded complete n-circular domain. Unless otherwise stated all the entire functions under consideration will be transcendental. We do not explain the standard definitions and notations in the theory of entire (and meromorphic) functions as those are available in [3]. Extending the notions of efinition 1 and efinition 2 we may give the following definitions: efinition 3 The hyper Gol dberg order (briefly hyper G-order) ρ f respect to the domain is defined by of f with ρ (f) ρ f = log [3] M f, (R). The hyper lower Gol dberg order (briefly hyper lower G-order) of f with respect to the domain is defined by λ (f) λ f = log [3] M f, (R). efinition 4 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin in C n. Then the relative hyper Gol dberg order ρ g, (f) of f with respect to g and the domain is defined as ρ g, (f) ρ f g, = log [2] M 1 g, (M f, (R)). Similarly the relative hyper lower order λ g, (f) of f with respect to g and the domain is defined by λ g, (f) λ f g, = log [2] M 1 g, (M f, (R)). Generalising our notion we may get that

4 1142 S. K. atta and M. Mallik efinition 5 The generalised Gol dberg order (briefly generalised G-order) ρ (k) (f) of f with respect to the domain is defined by ρ (k) (f) = log [k] M f, (R) where k =1, 2, 3,... The generalised lower Gol dberg order (briefly generalised lower G-order) of f with respect to the domain is defined by λ (k) (f) = log [k] M f, (R) where k =1, 2, 3,... efinition 6 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin in C n. Then the generalised relative Gol dberg order ρ (k) (f) of f with respect to g and the domain g, is defined by log [k] M 1 ρ (k) g, (f) = (M f, (R)), for k =1, 2, 3,... g, Similarly, the generalised relative lower Gol dberg order λ (k) (f) of f with respect to g and the domain is defined as g, log [k] M 1 λ (k) g, (f) = (M f, (R)), for k =1, 2, 3,... g, In the paper we establish some results on the comparative growth properties related to Gol dberg order (lower Gol dberg order) and relative Gol dberg order (relative lower Gol dberg order). 2 Theorems. In this section we present the main results of the paper. Theorem 1 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin in C n. Also let 0 <λ (fog) ρ (fog) < and 0 <λ (g) ρ (g) <. Then λ (fog) ρ (g) log [2] M g, (R) min {λ (fog) λ (g), ρ (fog) ρ (g) } max { λ (fog) λ (g), ρ (fog) } ρ (g) log [2] M g, (R) ρ (fog) λ (g).

5 Study of growth properties 1143 Proof. From the definition of Gol dberg order and lower Gol dberg order of an entire function g we have for arbitrary positive ε and for all sufficiently large values of R, log [2] M g, (R) (ρ (g)+ε) (1) and log [2] M g, (R) (λ (g) ε). (2) Also for a sequence of values of R, tending to infinity, log [2] M g, (R) (λ (g)+ε) (3) and log [2] M g, (R) (ρ (g) ε). (4) Again from the definition of Gol dberg order and lower Gol dberg order of the composite entire function fog we have for arbitrary positive ε and for all sufficiently large values of R, (ρ (fog)+ε) (5) and (λ (fog) ε). (6) Again for a sequence of values of R tending to infinity, (λ (fog)+ε) (7) and (ρ (fog) ε). (8) Now from (1) and (6) it follows for all sufficiently large values of R that log [2] M g, (R) As ε(> 0) is arbitrary we obtain that λ (fog) ε (g)+ε. ρ log [2] M g, (R) λ (fog) (g). (9) Again combining (2) and (7) we get for a sequence of values of R tending to infinity, log [2] M g, (R) ρ λ (fog)+ε λ (g) ε.

6 1144 S. K. atta and M. Mallik log [2] M g, (R) λ (fog) λ (g). (10) Similarly from (4) and (5) it follows for a sequence of values of R tending to infinity taht log [2] M g, (R) ρ (fog)+ε ρ (g) ε. log [2] M g, (R) ρ (fog) ρ (g). (11) Now combining (9), (10) and (11) we get that λ (fog) ρ (g) log [2] M g, (R) min {λ (fog) λ (g), ρ (fog) }. (12) ρ (g) Now from (3) and (6) we obtain for a sequence of values of R tending to infinity that log [2] M g, (R) Choosing ε 0 we get that λ (fog) ε λ (g)+ε. log [2] M g, (R) λ (fog) λ (g). (13) Again from (2) and (5) it follows for all sufficiently large values of R that log [2] M g, (R) ρ (fog)+ε λ (g) ε. log [2] M g, (R) ρ (fog) λ (g). (14) Similarly combining (1) and (8) we get for a sequence of values of R tending to infinity that log [2] M g, (R) ρ (fog) ε ρ (g)+ε.

7 Study of growth properties 1145 log [2] M g, (R) ρ (fog) ρ (g). (15) Therefore combining (13), (14) and (15) we get that max { λ (fog) λ (g), ρ (fog) } ρ (g) log [2] M g, (R) ρ (fog) λ (g). (16) Thus the theorem follows from (12) and (16). Remark 1 If we take 0 <λ (f) ρ (f) < instead of 0 <λ (g) ρ (g) < and the other conditions remain the same then also Theorem 1 holds with g replaced by f in the denominator as we see in the next theorem. Theorem 2 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin in C n. Also let 0 <λ (fog) ρ (fog) < and 0 <λ (f) ρ (f) <. Then λ (fog) ρ (f) log [2] M f, (R) min {λ (fog) λ (f), ρ (fog) ρ (f) } max { λ (fog) λ (f), ρ (fog) } ρ (f) log [2] M f, (R) ρ (fog) λ (f). Proof. From the definition of Gol dberg order and lower Gol dberg order of an entire function f we have for arbitrary positive ε and for all sufficiently large values of R, log [2] M f, (R) (ρ (f)+ε) (17) and log [2] M f, (R) (λ (f) ε). (18) Also for a sequence of values of R tending to infinity, log [2] M f, (R) (λ (f)+ε) (19) and log [2] M f, (R) (ρ (f) ε). (20)

8 1146 S. K. atta and M. Mallik Again from the definition of Gol dberg order and lower Gol dberg order of the composite entire function (or, composition of two entire functions f and g) fog we have for arbitrary positive ε and for all sufficiently large values of R, (ρ (fog)+ε) (21) and (λ (fog) ε). (22) Again for a sequence of values of R tending to infinity, (λ (fog)+ε) (23) and (ρ (fog) ε). (24) Now from (17) and (22) it follows for all sufficiently large values of R that log [2] M f, (R) λ (fog) ε ρ (f)+ε. log [2] M f, (R) λ (fog) ρ (f). (25) Again combining (18) and (23) we get for a sequence of values of R tending to infinity, log [2] M f, (R) λ (fog)+ε λ (f) ε. log [2] M f, (R) λ (fog) λ (f). (26) Similarly from (20) and (21) it follows for a sequence of values of R tending to infinity that log [2] M f, (R) ρ (fog)+ε ρ (f) ε. log [2] M f, (R) ρ (fog) ρ (f). (27)

9 Study of growth properties 1147 Now combining (25), (26) and (27) we get that λ (fog) ρ (f) log [2] M f, (R) min {λ (fog) λ (f), ρ (fog) }. (28) ρ (f) Now from (19) and (22) we obtain for a sequence of values of R tending to infinity, log [2] M f, (R) λ (fog) ε λ (f)+ε. Choosing ε(> 0) we get that log [2] M f, (R) λ (fog) λ (f). (29) Again from (18) and (21) it follows for all sufficiently large values of R, log [2] M f, (R) ρ (fog)+ε λ (f) ε. log [2] M f, (R) ρ (fog) λ (f). (30) Similarly combining (17) and (24) we get for a sequence of values of R tending to infinity log [2] M f, (R) ρ (fog) ε ρ (f)+ε. log [2] M f, (R) ρ (fog) ρ (f). (31) Therefore combining (29), (30) and (31) we get that max { λ (fog) λ (f), ρ (fog) } ρ (f) log [2] M f, (R) ρ (fog) λ (f). (32) Thus the theorem follows from (28) and (32). Extending the notion we may prove the subsequent theorems using hyper Gol dberg order.

10 1148 S. K. atta and M. Mallik Theorem 3 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin in C n. Also let 0 < λ (fog) ρ (fog) < and 0 < λ (g) ρ (g) <. Then λ (fog) ρ (g) log [3] M f, (R) min {λ (fog) λ (g), ρ (fog) ρ (g) } max { λ (fog) λ (g), ρ (fog) } ρ (g) log [3] M g, (R) ρ (fog) λ (g). Proof. From the definition of hyper Gol dberg order and hyper Gol dberg lower order of an entire function g we have for arbitrary positive ε and for all sufficiently large values of R, log [3] M g, (R) (ρ (g)+ε) (33) and log [3] M g, (R) (λ (g) ε). (34) Also for the sequence of values of R tending to infinity, log [3] M g, (R) (λ (g)+ε) (35) and log [3] M g, (R) (ρ (g) ε). (36) Again from the definition of hyper Gol dberg order and hyper lower Gol dberg order of the copmposite entire function fog we have for arbitrary positive ε and for all sufficiently large values of R, (ρ (fog)+ε) (37) and (λ (fog) ε). (38) Again for a sequence of values of R tending to infinity, (λ (fog)+ε) (39) and (ρ (fog) ε). (40)

11 Study of growth properties 1149 Now from (33) and (38) it follows for all sufficiently large values of R that log [3] M g, (R) λ (fog) ε ρ (g)+ε. As ε(> 0) is arbitrary we obtain that log [3] M g, (R) λ (fog) ρ (g). (41) Again combining (34) and (39) we get for a sequence of values of R tending to infinity, log [3] M g, (R) λ (fog)+ε λ (g) ε. log [3] M g, (R) λ (fog) λ (g). (42) Similarly from (36) and (37) it follows for a sequence of values of R tending to infinity that log [3] M g, (R) ρ (fog)+ε ρ (g) ε. log [3] M g, (R) ρ (fog) ρ (g). (43) Now combining (41), (42) and (43) we get that λ (fog) ρ (g) log [3] M g, (R) min {λ (fog) λ (g), ρ (fog) }. (44) ρ (g) Now from (35) and (38) we obtain for a sequence of values of R tending to infinity that log [3] M g, (R) λ (fog) ε λ (g)+ε.

12 1150 S. K. atta and M. Mallik Choosing ε 0 we get that log [3] M g, (R) λ (fog) λ (g). (45) Again from (34) and (37) it follows for all sufficiently large values of R, log [3] M g, (R) ρ (fog)+ε λ (g) ε. log [3] M g, (R) ρ (fog) λ (g). (46) Similarly combining (33) and (40) we get for a sequence of values of R tending to infinity that log [3] M g, (R) ρ (fog) ε ρ (g)+ε. log [3] M g, (R) ρ (fog) ρ (g). (47) Therefore combining (45), (46) and (47) we get that max { λ (fog) λ (g), ρ (fog) } ρ (g) log [3] M g, (R) ρ (fog) λ (g). (48) Thus the theorem follows from (44) and (48). Remark 2 If we take 0 < λ (f) ρ (f) < instead of 0 < λ (g) ρ (g) < and the other conditions remain the same then also Theorem 3 holds with g replaced by f in the denominator as we see in the next theorem. Theorem 4 Let f and g be two entire functions of n variables and be a bounded complete n-circular domain with centre at origin an C n. Also let 0 < λ (fog) ρ (fog) < and 0 < λ (f) ρ (f) <. Then λ (fog) ρ (f) log [3] M g, (R) min {λ (fog) λ (f), ρ (fog) ρ (f) } max { λ (fog) λ (f), ρ (fog) } ρ (f) log [3] M f, (R) ρ (fog) λ (f).

13 Study of growth properties 1151 Proof. From the definition of hyper Gol dberg order and hyper lower Gol dberg order of an entire function f we have for arbitrary positive ε and for all sufficiently large values of R, log [3] M f, (R) (ρ (f)+ε) (49) and log [3] M f, (R) (λ (f) ε). (50) Also for a sequence of values of R tending to infinity, log [3] M f, (R) (λ (f)+ε) (51) and log [3] M f, (R) (ρ (f) ε). (52) Again from the definition of hyper Gol dberg order and hyper lower Gol dberg order of the composite entire function fog we have for arbitrary positive ε and for all sufficiently large values of R, (ρ (fog)+ε) (53) and (λ (fog) ε). (54) Again for a sequence of values of R tending to infinity, (λ (fog)+ε) (55) and (ρ (fog) ε). (56) Now from (49) and (54) it follows for all sufficiently large values of R that log [3] M f, (R) λ (fog) ε ρ (f)+ε. As ε(> 0) is arbitrary we obtain that log [3] M f, (R) λ (fog) ρ (f). (57) Again combining (50) and (55) we get for a sequence of values of R tending to infinity, log [3] M f, (R) λ (fog)+ε λ (f) ε.

14 1152 S. K. atta and M. Mallik log [3] M f, (R) λ (fog) λ (f). (58) Similarly from (52) and (53) it follows for a sequence of values of R tending to infinity that log [3] M f, (R) ρ (fog)+ε ρ (f) ε. log [3] M f, (R) ρ (fog) ρ (f). (59) Now combining (57), (58) and (59) we get that λ (fog) ρ (f) log [3] M f, (R) min {λ (fog) λ (f), ρ (fog) ρ (f) } (60) Now from (51) and (54) we obtain for a sequence of values of R tending to infinity, log [3] M f, (R) λ (fog) ε λ (f)+ε. Choosing ε 0 we get that log [3] M f, (R) λ (fog) λ (f). (61) Again from (50) and (53) it follows for all sufficiently large values of R, log [3] M f, (R) ρ (fog)+ε λ (f) ε. log [3] M f, (R) ρ (fog) λ (f). (62) Similarly combining (49) and (56) we get for a sequence of values of R tending to infinity that log [3] M f, (R) ρ (fog) ε ρ (f)+ε.

15 Study of growth properties 1153 log [3] M f, (R) ρ (fog) ρ (f). (63) Therefore combining (61), (62) and (63) we get that max { λ (fog) λ (f), ρ (fog) } ρ (f) log [3] M f, (R) ρ (fog) λ (f). (64) Thus the theorem follows from (60) and (64). References [1] Mondal.B.C. and Roy.C.: Relative Gol, dberg order of an entire function of several variables, Bull. Cal. Math. Soc,Vol. 102, No.4 (2010), pp [2] Fuks.B.A.: Introduction to the Theory of Analytic functions of several variables, 1965 [3] Valiron.G:Lectures on the general theory of integral functions, Chelsea Publishing Company, 1949 Received: October, 2010