Some inequalities in connection to relative orders of entire functions of several complex variables

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1 Int. J. Nonlinear Anal. Appl. 7 (2016) No. 2, ISSN: (electronic) Some inequalities in connection to relative orders of entire functions of several complex variables Sanjib Kumar Datta a,, Tanmay Biswas b, Debasmita Dutta c a Department of Mathematics, University of Kalyani, P.O. Kalyani, Dist-Nadia,PIN , West Bengal, India b Rajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar, Dist-Nadia,PIN , West Bengal, India c Mohanpara Nibedita Balika Vidyalaya (High),P.o - Amrity, Block - English Bazar, Dist.- District - Malda, PIN , West Bengal, India (Communicated by A. Ebadian) Abstract Let f, g h be all entire functions of several complex variables. In this paper we would like to establish some inequalities on the basis of relative order relative lower order of f with respect to g when the relative orders relative lower orders of both f g with respect to h are given. Keywords: entire function; several complex variables; relative order; relative lower order MSC: Primary 32A15; Secondary 30D Introduction preliminaries Let f be an entire function of two complex variables holomorphic in the closed polydisc U = {(z 1, z 2 ) : z i r i, i = 1, 2 for all r 1 0, r 2 0} M f (r 1, r 2 ) = max { f (z 1, z 2 ) : z i r i, i = 1, 2}. Then in view of maximum principal Hartogs theorem {[7], p. 2, p. 51}, M f (r 1, r 2 ) is an increasing functions of r 1, r 2. The following definition is well known: Definition 1.1. {[7], p. 339, (see also [1])} The order v2 ρ f the lower order v2 λ f of an entire function f of two complex variables are defined as log [2] M f (r 1, r 2 ) v 2 ρ f = r 1,r 2 log (r 1 r 2 ) log [2] M f (r 1, r 2 ) v2 λ f = lim inf, r 1,r 2 log (r 1 r 2 ) Corresponding author addresses: sanjib_kr_datta@yahoo.co.in (Sanjib Kumar Datta), tanmaybiswas_math@rediffmail.com (Tanmay Biswas), debasmita.dut@gmail.com (Debasmita Dutta) Received: July 2015 Revised: October 2016

2 134 Datta, Biswas, Dutta ( ) where log [k] x = log log [k 1] x, k = 1, 2, 3,... log [0] x = x. If we consider the above definiton for single variable, then the definition coincides with the classical definition of order (see [14]) which is as follows: Definition 1.2. ([14]) The order ρ f the lower order λ f of an entire function f are defined in the following way: log [2] M f (r) ρ f = r log r where M f (r) = max { f (z) : z = r}. λ f = lim inf r log [2] M f (r), log r If f is non-constant then M f (r) is strictly increasing continuous, its inverse f : ( f (0), ) (0, ) exists is such that lim f (s) =. Bernal {[2], [3]} introduced the s definition of relative order of g with respect to f, denoted by ρ f (g) as follows : ρ g (f) = inf {µ > 0 : M f (r) < (r µ ) for all r > r 0 (µ) > 0} = 1 M f (r). r log r The definition coincides with the classical one [14] if g (z) = exp z. During the past decades, several authors ( see [5],[9],[10],[11],[12],[13]) made close investigations on the properties of relative order of entire functions of single variable. In the case of relative order, it was then natural for Banerjee Dutta [4] to define the relative order of entire functions of two complex variables as follows: Definition 1.3. ([4]) The relative order between two entire functions of two complex variables denoted by v2 ρ g (f) is defined as: v 2 ρ g (f) = inf {µ > 0 : M f (r 1, r 2 ) < (r µ 1, r µ 2 ) ; r 1 R (µ), r 2 R (µ)} = 1 M f (r 1, r 2 ) r 1,r 2 log (r 1 r 2 ) where f g are entire functions holomorphic in the closed polydisc U = {(z 1, z 2 ) : z i r i, i = 1, 2 for all r 1 0, r 2 0} the definition coincides with Definition 1.1 {see [4]} if g (z) = exp (z 1 z 2 ). Extending this notion, Dutta [6] introduced the idea of relative order of entire functions of several complex variables in the following way: Definition 1.4. ([6]) Let f(z 1, z 2,..., z n ) g(z 1, z 2,..., z n ) be any two entire functions of n complex variables z 1, z 2,..., z n with maximum modulus functions M f (r 1, r 2,..., r n ) (r 1, r 2,..., r n ) respectively then the relative order of f with respect to g, denoted by vn ρ g (f) is defined by ρ g (f) = inf {µ > 0 : M f (r 1, r 2,..., r n ) < (r µ 1, r µ 2,..., r µ n) ; for r i R (µ), i = 1, 2,..., n}.

3 Some inequalities in connection to relative orders... 7 (2016) No. 2, The above definition can equivalently be written as 1 M f (r 1, r 2,..., r n ) ρ g (f) =. r 1,r 2,...,r n Similarly, one can define the relative lower order of f with respect to g denoted by vn λ g (f) as follows: 1 M f (r 1, r 2,..., r n ) λ g (f) = lim inf. r 1,r 2,...,r n If we consider g(z 1, z 2,..., z n ) = exp (z 1 z 2...z n ), then Definition 1.4 reduces to the following classical definition of order lower order in connection with several complex variables: Definition 1.5. The order vn ρ f the lower order vn λ f of an entire function f of two complex variables are defined as log [2] M f (r 1, r 2,..., r n ) ρ f = r 1,r 2,...,r n log [2] M f (r 1, r 2,..., r n ) λ f = lim inf. r 1,r 2,...,r n Also an entire function of several complex variables for which order lower order are the same is said to be of regular growth. The function exp (z 1 z 2...z n ) is an example of regular growth of entire function of several complex variables. Further the functions which are not of regular growth are said to be of irregular growth. Similarly for an entire function of several complex variables for which relative order relative lower order with respect to another entire function of several complex variables are the same is said to be of regular relative growth with respect to that entire function. Also the functions which are not of regular relative growth with respect to entire functions are said to be of irregular relative growth with respect to respective entire functions. Now a question may arise about relative order (relative lower order) of f with respect to another entire function g when relative order (relative lower oreder) of f g with respect to another entire function h are respectively given. In this paper we intend to provide this answer. Interested researchers may also think over to establish such type of results on the coupled systems of equations with entire polynomial functions {cf. [8]}. We do not explain the stard definitions notations in the theory of entire function of two complex variables as those are available in [7]. 2. The main results In this section we present the main results of the paper. Theorem 2.1. Let f, g h be any three entire functions of several complex variables such that relative order (relative lower order) of f with respect to h relative order (relative lower order) of f with respect to h are vn ρ h (f) ( vn λ h (f)) vn ( vn λ h (g)) respectively. Then { } λ h (f) v λ g (f) min n λ h (f) λ h (g), ρ { max λ h (f) λ h (g), ρ } vn ρ g (f) vn ρ h (f) λ h (g).

4 136 Datta, Biswas, Dutta Proof. From the definitions of vn ρ h (f) vn λ h (f), we have for all sufficiently large values of r 1, r 2,..., r n that h M f (r 1, r 2,..., r n ) exp {( vn ρ h (f) + ε) } M f (r 1, r 2,..., r n ) M h {( vn ρ h (f) + ε) }], (2.1) h M f (r 1, r 2,..., r n ) exp {( vn λ h (f) ε) } M f (r 1, r 2,..., r n ) M h {( vn λ h (f) ε) }], (2.2) also for a sequence of values of r 1, r 2,..., r n tending to infinity we get that h M f (r 1, r 2,..., r n ) exp {( vn ρ h (f) ε) } M f (r 1, r 2,..., r n ) M h {( vn ρ h (f) ε) }], (2.3) h M f (r 1, r 2,..., r n ) exp {( vn λ h (f) + ε) } M f (r 1, r 2,..., r n ) M h {( vn λ h (f) + ε) }]. (2.4) Similarly from the definitions of vn vn λ h (g), it follows for all sufficiently large values of r 1, r 2 that h (r 1, r 2,..., r n ) exp {( vn + ε) } (r 1, r 2,..., r n ) M h {( vn + ε) }] M h (r 1, r 2,..., r n ) [ log (r1 r 2...r n ) ( vn + ε) ]]. (2.5) h (r 1, r 2,..., r n ) exp {( vn λ h (g) ε) } (r 1, r 2,..., r n ) M h {( vn λ h (g) ε) }] log (r1 r 2...r n ) M h (r 1, r 2,..., r n ) ( vn λ h (g) ε) for a sequence of values of r 1, r 2,..., r n tending to infinity we obtain that (2.6) h (r 1, r 2,..., r n ) exp {( vn ε) } (r 1, r 2,..., r n ) M h {( vn ε) }] M h (r 1, r 2,..., r n ) [ log (r1 r 2...r n ) ( vn ε) ]]. (2.7) h (r 1, r 2,..., r n ) exp {( vn λ h (g) + ε) } (r 1, r 2,..., r n ) M h {( vn λ h (g) + ε) }] M h (r 1, r 2,..., r n ) [ log (r1 r 2...r n ) ( vn λ h (g) + ε) ]]. (2.8) Now from (2.3) in view of (2.5), we get for a sequence of values of r 1, r 2,..., r n tending to infinity that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn ρ h (f) ε) }]

5 Some inequalities in connection to relative orders... 7 (2016) No. 2, M f (r 1, r 2,..., r n ) 1 vn ρ h (f) ε) } ( vn + ε) 1 M f (r 1, r 2,..., r n ) ( ρ h (f) ε) ( vn + ε) log (r 1r 2...r n ) log As ε (> 0) is arbitrary, it follows that ( ρ h (f) ε) ( vn + ε). 1 M f (r 1, r 2,..., r n ) ρ r 1,r 2 ρ vn ρ g (f). (2.9) Analogously, from (2.2) in view of (2.8) it follows for a sequence of values of r 1, r 2,..., r n tending to infinity that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn λ h (f) ε) }] 1 M f (r 1, r 2,..., r n ) 1 vn λ h (f) ε) } ( vn λ h (g) + ε) 1 M f (r 1, r 2,..., r n ) ( λ h (f) ε) ( vn λ h (g) + ε) log (r 1r 2...r n ) log ( λ h (f) ε) ( vn λ h (g) + ε). Since ε (> 0) is arbitrary, we get from above that 1 M f (r 1, r 2,..., r n ) λ r 1,r 2 λ h (g) λ vn ρ g (f) λ h (g). (2.10) Again in view of (2.6), we have from (2.1) for all sufficiently large values of r 1, r 2,..., r n that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn ρ h (f) + ε) }] 1 M f (r 1, r 2,..., r n ) 1 vn ρ h (f) + ε) } ( vn λ h (g) ε)

6 138 Datta, Biswas, Dutta 1 M f (r 1, r 2,..., r n ) ( ρ h (f) + ε) ( vn λ h (g) ε) log (r 1r 2...r n ) log Since ε (> 0) is arbitrary, we obtain that ( ρ h (f) + ε) ( vn λ h (g) ε). 1 M f (r 1, r 2,..., r n ) ρ r 1,r 2 λ h (g) ρ vn ρ g (f) λ h (g). (2.11) Again from (2.2) in view of (2.5), we get for all sufficiently large values of r 1, r 2,..., r n that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn λ h (f) ε) }] 1 M f (r 1, r 2,..., r n ) 1 vn λ h (f) ε) } ( vn + ε) 1 M f (r 1, r 2,..., r n ) ( λ h (f) ε) ( vn + ε) log (r 1r 2...r n ) log ( λ h (f) ε) ( vn + ε). As ε (> 0) is arbitrary, it follows from above that 1 M f (r 1, r 2,..., r n ) λ lim inf r 1,r 2 λ vn λ g (f). (2.12) Also in view of (2.7), we get from (2.1) for a sequence of values of r 1, r 2,..., r n tending to infinity that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn ρ h (f) + ε) }] 1 M f (r 1, r 2,..., r n ) 1 vn ρ h (f) + ε) } ( vn ε) 1 M f (r 1, r 2,..., r n ) ( ρ h (f) + ε) ( vn ε) log (r 1r 2...r n ) log ( ρ h (f) + ε) ( vn ε).

7 Some inequalities in connection to relative orders... 7 (2016) No. 2, Since ε (> 0) is arbitrary, we get from above that 1 M f (r 1, r 2,..., r n ) ρ lim inf r 1,r 2 ρ vn λ g (f). (2.13) Similarly from (2.4) in view of (2.6), it follows for a sequence of values of r 1, r 2,..., r n tending to infinity that 1 M f (r 1, r 2,..., r n ) 1 M h {( vn λ h (f) + ε) }] 1 M f (r 1, r 2,..., r n ) 1 vn λ h (f) + ε) } ( vn λ h (g) ε) 1 M f (r 1, r 2,..., r n ) ( λ h (f) + ε) ( vn λ h (g) ε) log (r 1r 2...r n ) log ( λ h (f) + ε) ( vn λ h (g) ε). As ε (> 0) is arbitrary, we obtain from above that 1 M f (r 1, r 2,..., r n ) λ lim inf r 1,r 2 log (r 1 r 2r1 r 2...r n ) λ h (g) λ vn λ g (f) λ h (g). (2.14) Thus the theorem follows from (2.9), (2.10), (2.11), (2.12), (2.13) (2.14). In view of Theorem 2.1, one can easily verify the following corollaries: Corollary 2.2. Let f be an entire function of several complex variables with regular relative growth with respect to an entire function h of several complex variables g be entire another entire function of several complex variables. Then λ g (f) = vn ρ h (f) In addition, if vn ρ h (f) = vn, then λ g (f) = vn ρ f (g) = 1. vn ρ g (f) = vn ρ h (f) λ h (g). Corollary 2.3. Let f, g, h be three entire functions of several complex variables such that g is of regular relative growth with respect to an entire function h. Then λ g (f) = vn λ h (f) In addition, if vn ρ h (f) = vn then ρ g (f) = vn λ f (g) = 1. vn ρ g (f) = vn ρ h (f).

8 140 Datta, Biswas, Dutta Corollary 2.4. Let f g be any two entire functions of several complex variables with regular relative growth with respect to another entire function h of several complex variables respectively. Then λ g (f) = vn ρ g (f) = vn ρ h (f). Corollary 2.5. Let f g be any two entire functions of several complex variables with regular relative growth regular relative growth with respect to another entire function h of several complex variables respectively. Also suppose that vn ρ h (f) = vn. Then λ g (f) = vn ρ g (f) = vn λ f (g) = vn ρ f (g) = 1. Corollary 2.6. Let f, g h be any three entire functions of several complex variables such that either f is not of regular relative growth or g is not of regular relative growth with respect to h. Then ρ g (f) vn ρ f (g) 1 when f g are both of regular relative growth with respect to h respectively, then ρ g (f) vn ρ f (g) = 1. Corollary 2.7. Let f, g h be any three entire functions of several complex variables such that either f is not of regular relative growth or g is not of regular relative growth with respect to h. Then λ g (f) vn λ f (g) 1. when f g are both of regular relative growth with respect to h respectively, then λ g (f) vn λ f (g) = 1. Corollary 2.8. Let f g be any two entire functions of several complex variables. Then (i) vn λ g (f) = when vn = 0, (ii) vn ρ g (f) = when vn λ h (g) = 0, (iii) vn λ g (f) = 0 when vn = (iv) vn ρ g (f) = 0 when vn λ h (g) =. Corollary 2.9. Let f g be any two entire functions of several complex variables. Then (i) vn ρ g (f) = 0 when vn ρ h (f) = 0, (ii) vn λ g (f) = 0 when vn λ h (f) = 0, (iii) vn ρ g (f) = when vn ρ h (f) = (iv) vn λ g (f) = when vn λ h (f) =. Acknowledgment The authors are thankful to the referees for their valuable suggestions which have considerably improved the paper.

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