Some results relating to (p,q)-th relative Gol'dberg order and (p,q)-relative Gol'dberg type of entire functions of several variables
aa r X i v : . [ m a t h . C V ] F e b SOME RESULTS RELATING TO ( p, q ) -TH RELATIVE GOL’DBERGORDER AND ( p, q ) -RELATIVE GOL’DBERG TYPE OF ENTIREFUNCTIONS OF SEVERAL VARIABLES TANMAY BISWAS
Abstract.
In this paper we introduce the notions of ( p, q ) -th relative Gol’dbergorder and ( p, q ) -th relative Gol’dberg type of entire functions of several com-plex variables where p, q are any positive integers. Then we study some growthproperties of entire functions of several complex variables on the basis of their ( p, q ) -th relative Gol’dberg order and ( p, q ) -th relative Gol’dberg type. Introduction and Definitions.
Let C n and R n respectively denote the complex and real n -space . Also let usindicate the point ( z , z , · · · , z n ) , ( m , m , · · · , m n ) of C n or I n by their correspondingunsuffixed symbols z, m respectively where I denotes the set of non-negative integers.The modulus of z , denoted by | z | , is defined as | z | = (cid:16) | z | + · · · + | z n | (cid:17) . If thecoordinates of the vector m are non-negative integers, then z m will denote z m · · · z m n n and k m k = m + · · · + m n .If D ⊆ C n ( C n denote the n -dimensional complex space ) be an arbitrary boundedcomplex n -circular domain with center at the origin of coordinates then for any entirefunction f ( z ) of n complex variables and R > , M f,D ( R ) may be define as M f,D ( R ) =sup z ∈ D R | f ( z ) | where a point z ∈ D R if and only if zR ∈ D. If f ( z ) is non-constant, then M f,D ( R ) is strictly increasing and its inverse M − f,D : ( | f (0) | , ∞ ) → (0 , ∞ ) exists suchthat lim R →∞ M − f,D ( R ) = ∞ . Considering this, the
Gol’dberg order (resp.
Gol’dberg lower order ) { cf. [4], [5] } of an entire function f ( z ) with respect to any bounded complete n -circular domain D isgiven by ρ f,D = lim R → + ∞ log [2] M f,D ( R )log R (resp. λ f,D = lim R → + ∞ log [2] M f,D ( R )log R ) . where log [ k ] R = log (cid:16) log [ k − R (cid:17) for k = 1 , , , · · · ; log [0] R = R and exp [ k ] R =exp (cid:0) exp [ k − R (cid:1) for k = 1 , , , · · · ; exp [0] R = R. Key words and phrases. ( p, q ) -th relative Gol’dberg order, ( p, q ) -th relative Gol’dberg lower order, ( p, q ) -th relative Gol’dberg type, ( p, q ) -th relative Gol’dberg weak type, growth. AMS Subject Classification (2010) : 30 D , D It is well known that ρ f,D is independent of the choice of the domain D , andtherefore we write ρ f instead of ρ f,D (resp. λ f instead of λ f,D ) { cf. [4], [5] } . For any bounded complete n -circular domain D, an entire function of n -complexvariables for which Gol’dberg order and
Gol’dberg lower order are the same is said to beof regular growth . Functions which are not of regular growth are said to be of irregulargrowth .To compare the relative growth of two entire functions of n -complex variableshaving same non zero finite Gol’dberg order , one may introduce the definition of
Gol’dbergtype and
Gol’dberg lower type in the following manner:
Definition 1. { cf. [4] , [5] } The
Gol’dberg type and
Gol’dberg lower type respectivelydenoted by ∆ f,D and ∆ f,D of an entire function f ( z ) of n -complex variables with respectto any bounded complete n -circular domain D are defined as follows: ∆ f,D = lim R → + ∞ log M f,D ( R )( R ) ρ f and ∆ f,D = lim R → + ∞ log M f,D ( R )( R ) ρ f , < ρ f < + ∞ . Analogously to determine the relative growth of two entire functions of n -complexvariables having same non zero finite Gol’dberg lower order, one may introduce thedefinition of
Gol’dberg weak type in the following way:
Definition 2.
The
Gol’dberg weak type denoted by τ f,D of an entire function f ( z ) of n -complex variables with respect to any bounded complete n -circular domain D isdefined as follows: τ f,D = lim R → + ∞ log M f,D ( R )( R ) λ f , < λ f < + ∞ . Also one may define the growth indicator τ f,D in the following manner : τ f,D = lim R → + ∞ log M f,D ( R )( R ) λ f , < λ f < + ∞ Gol’dberg has shown that [5]
Gol’dberg type depends on the domain D. Hence allthe growth indicators define in Definition 1 and Definition 2 are also depend on D. However, extending the notion of
Gol’dberg order , Datta and Maji [1] defined theconcept of ( p, q ) -th Gol’dberg order (resp. ( p, q ) -th Gol’dberg lower order ) of an entirefunction f ( z ) for any bounded complete n -circular domain D where p ≥ q ≥ ρ f,D ( p, q ) = lim R → + ∞ log [ p ] M f,D ( R )log [ q ] R = lim R → + ∞ log [ p ] R log [ q ] M − f,D ( R )(resp. λ f,D ( p, q ) = lim R → + ∞ log [ p ] M f,D ( R )log [ q ] R = lim R → + ∞ log [ p ] R log [ q ] M − f,D ( R ) ) . These definitions extended the generalized Gol’dberg order ρ [ l ] f,D (resp. generalizedGol’dberg lower order λ [ l ] f,D ) of an entire function f ( z ) for any bounded complete n -circular domain D for each integer l ≥ OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 3 ρ [ l ] f,D = ρ f,D ( l,
1) (resp. λ [ l ] f,D = λ f,D ( l, . Clearly ρ f,D (2 ,
1) = ρ f,D (resp. λ f,D (2 ,
1) = λ f,D ) . Further in the line of Gol’dberg { cf. [4], [5] } , one can easily verify that ρ f,D ( p, q )(resp. λ f,D ( p, q )) is independent of the choice of the domain D , and therefore one canwrite ρ f ( p, q ) (resp. λ f ( p, q )) instead of ρ f,D ( p, q ) (resp. λ f,D ( p, q )).In this connection let us recall that if 0 < ρ f ( p, q ) < ∞ , then the followingproperties hold ρ f ( p − n, q ) = ∞ for n < p, ρ f ( p, q − n ) = 0 for n < q , and ρ f ( p + n, q + n ) = 1 for n = 1 , , ... Similarly for 0 < λ f ( p, q ) < ∞ , one can easily verify that λ f ( p − n, q ) = ∞ for n < p, λ f ( p, q − n ) = 0 for n < q , and λ f ( p + n, q + n ) = 1 for n = 1 , , .... . Recalling that for any pair of integer numbers m, n the Kroenecker function isdefined by δ m,n = 1 for m = n and δ m,n = 0 for m = n , the aforementioned propertiesprovide the following definition. Definition 3.
For any bounded complete n -circular domain D, an entire function f ( z ) of n -complex variables is said to have index-pair (1 , if < ρ f (1 , < ∞ . Otherwise, f ( z ) is said to have index-pair ( p, q ) = (1 , , p ≥ q ≥ , if δ p − q, < ρ f ( p, q ) < ∞ and ρ f ( p − , q − / ∈ R + . Definition 4.
For any bounded complete n -circular domain D, an entire function f ( z ) of n -complex variables is said to have lower index-pair (1 , if < λ f (1 , < ∞ .Otherwise, f ( z ) is said to have lower index-pair ( p, q ) = (1 , , p ≥ q ≥ , if δ p − q, <λ f ( p, q ) < ∞ and λ f ( p − , q − / ∈ R + . For any bounded complete n -circular domain D, an entire function f ( z ) of n -complex variables of index-pair ( p, q ) is said to be of regular ( p, q ) - Gol’dberg growth ifits ( p, q ) -th Gol’dberg order coincides with its ( p, q ) -th Gol’dberg lower order , otherwise f ( z ) is said to be of irregular ( p, q ) -Gol’dberg growth .To compare the relative growth of two entire functions having same non zerofinite ( p, q ) -Gol’dberg order , one may introduce the definition of ( p, q ) -Gol’dberg type and ( p, q )- Gol’dberg lower type in the following manner:
Definition 5.
The ( p, q )-th Gol’dberg type and ( p, q ) -th Gol’dberg lower type respec-tively denoted by ∆ f,D ( p, q ) and ∆ f,D ( p, q ) of an entire function f ( z ) of n -complexvariables with respect to any bounded complete n -circular domain D are defined as fol-lows: ∆ f,D ( p, q ) = lim R → + ∞ log [ p − M f,D ( R ) h log [ q − R i ρ f ( p,q ) and ∆ f,D ( p, q ) = lim R → + ∞ log [ p − M f,D ( R ) h log [ q − R i ρ f ( p,q ) , < ρ f ( p, q ) < + ∞ , where p ≥ q ≥ . TANMAY BISWAS
An entire function f ( z ) of n -complex variables of index-pair ( p, q ) is said to beof perfectly regular ( p, q ) -Gol’dberg growth if its ( p, q ) -th Gol’dberg order coincides withits ( p, q ) - th Gol’dberg lower order as well as its ( p, q ) th Gol’dberg type coincides withits ( p, q ) th Gol’dberg lower type .Analogously to determine the relative growth of two entire functions of n -complexvariables having same non zero finite ( p, q )- Gol’dberg lower order, one may introduce thedefinition of ( p, q )- Gol’dberg weak type in the following way:
Definition 6.
The ( p, q ) - th Gol’dberg weak type denoted by τ f,D ( p, q ) of an entirefunction f ( z ) of n -complex variables with respect to any bounded complete n -circulardomain D is defined as follows: τ f,D ( p, q ) = lim R → + ∞ log [ p − M f,D ( R ) h log [ q − R i λ f ( p,q ) , < λ f ( p, q ) < + ∞ . Also one may define the growth indicator τ f,D ( p, q ) in the following manner : τ f.D ( p, q ) = lim R → + ∞ log [ p − M f ( R ) h log [ q − R i λ f ( p,q ) , < λ f ( p, q ) < + ∞ , where p ≥ q ≥ . Definition 5 and Definition 6 are extended the generalized Gol’dberg type ∆ [ l ] f,D (resp. generalized Gol’dberg lower type ∆ [ l ] f,D ) and generalized Gol’dberg weak order τ [ l ] f,D of an entire function f ( z ) of n -complex variables with respect to any bounded complete n -circular domain D for each integer l ≥ [ l ] f,D = ∆ f,D ( l,
1) (resp. ∆ [ l ] f,D = ∆ f ( l, τ [ l ] f,D = τ f ( l,
1) (resp. τ [ l ] f,D = τ f ( l, . Clearly ∆ f,D (2 ,
1) = ∆ f,D (resp. ∆ f,D (2 ,
1) = ∆ f,D ) and τ f,D (2 ,
1) = τ f,D (resp. τ f,D (2 ,
1) = τ f,D ) . Since Gol’dberg has shown that [5]
Gol’dberg type depends on the domain D, therefore all the growth indicators define in Definition 5 and Definition 6 are also dependon D. For any two entire functions f ( z ) and g ( z ) of n -complex variables and for anybounded complete n -circular domain D with center at all the origin C n , Mondal andRoy [6] introduced the concept relative Gol’dberg order which is as follows: ρ g,D ( f ) = inf { µ > M f,D ( R ) < M g,D ( R µ ) for all R > R ( µ ) > } = lim R → + ∞ log M − g,D M f,D ( R )log R .
In [6], Mandal and Roy also proved that the relative Gol’dberg order of f ( z ) withrespect to g ( z ) is independent of the choice of the domain D . So the relative Gol’dbergorder of f ( z ) with respect to g ( z ) may be denoted as ρ g ( f ) instead of ρ g,D ( f ) . OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 5
Likewise, one can define the relative Gol’dberg lower order λ g,D ( f ) in the followingmanner: λ g,D ( f ) = lim R → + ∞ log M − g,D M f,D ( R )log R .
In the line of Mandal and Roy { cf. [6] } , one can also verify that λ g,D ( f ) isindependent of the choice of the domain D , and therefore one can write λ g ( f ) insteadof λ g,D ( f ) . In the case of relative Gol’dberg order, it therefore seems reasonable to definesuitably the ( p, q ) -th relative Gol’dberg order of entire function of n -complex variables and for any bounded complete n -circular domain D with center at the origin in C n . Withthis in view one may introduce the definition of ( p, q ) -th relative Gol’dberg order ρ ( p,q ) g,D ( f )of an entire function f ( z ) with respect to another entire function g ( z ) where both f ( z )and g ( z ) are of n -complex variables and D be any bounded complete n -circular domainwith center at the origin in C n , in the light of index-pair. Our next definition avoidsthe restriction p > q and gives the more natural particular case of Generalized Gol’dbergorder i.e, ρ [ l, g,D ( f ) = ρ [ l ] g,D ( f ). Definition 7.
Let f ( z ) and g ( z ) be any two entire functions of n -complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where p, q, m are positive integers suchthat m ≥ q ≥ and m ≥ p ≥ and D be any bounded complete n -circular domain withcenter at the origin in C n . Then the ( p, q ) -th relative Gol’dberg order of f ( z ) with respectto g ( z ) is defined as ρ ( p,q ) g,D ( f ) = inf ( µ > M f,D ( r ) < M g,D (cid:16) exp [ p ] (cid:16) µ log [ q ] R (cid:17)(cid:17) for all R > R ( µ ) > ) = lim R → + ∞ log [ p ] M − g,D M f,D ( R )log [ q ] R = lim R → + ∞ log [ p ] M − g,D ( R )log [ q ] M − f,D ( R ) . Similarly, the ( p, q ) -th relative Gol’dberg lower order of f ( z ) with respect to g ( z ) is defined by: λ ( p,q ) g,D ( f ) = lim R → + ∞ log [ p ] M − g.D M f,D ( R )log [ q ] R = lim R → + ∞ log [ p ] M − g,D ( R )log [ q ] M − f,D ( R ) . In this connection, one may introduce the definition of relative index-pair of anentire function with respect to another entire function (both of n -complex variables ) which is relevant in the sequel : Definition 8. [ ? ] Let f ( z ) and g ( z ) be any two entire functions (both of n -complexvariables ) with index-pairs ( m, q ) and ( m, p ) respectively where m ≥ q ≥ , m ≥ p ≥ and D be any bounded complete n -circular domain. Then the entire function f ( z ) issaid to have relative index-pair ( p, q ) with respect to another entire function g ( z ) , if b < ρ ( p,q ) g,D ( f ) < ∞ and ρ ( p − ,q − g,D ( f ) is not a nonzero finite number, where b = 1 if p = q = m and b = 0 for otherwise. Moreover if < ρ ( p,q ) g,D ( f ) < ∞ , then ρ ( p − n,q ) g,D ( f ) = ∞ for n < p, ρ ( p,q − n ) g,D ( f ) = 0 for n < q and TANMAY BISWAS ρ ( p + n,q + n ) g,D ( f ) = 1 for n = 1 , , .... . Similarly for < λ ( p,q ) g,D ( f ) < ∞ , one can easily verify that λ ( p − n,q ) g,D ( f ) = ∞ for n < p, λ ( p,q − n ) g,D ( f ) = 0 for n < q and λ ( p + n,q + n ) g,D ( f ) = 1 for n = 1 , , .... . Further an entire function f ( z ) for which ( p, q ) -th relative Gol’dberg order and( p, q ) -th relative Gol’dberg lower order with respect to another entire function g ( z ) arethe same is called a function of regular relative ( p, q ) - Gol’dberg growth with respect to g ( z ). Otherwise, f ( z ) is said to be irregular relative ( p, q ) -Gol’dberg growth .with respectto g ( z ).Next we introduce the definition of ( p, q ) -th relative Gol’dberg type and ( p, q )-th relative Gol’dberg lower type in order to compare the relative growth of two entirefunctions of n -complex variables having same non zero finite ( p, q ) -th relative Gol’dbergorder with respect to another entire function of n -complex variables. Definition 9.
Let f ( z ) and g ( z ) be any two entire functions of n -complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where p, q, m are positive integers suchthat m ≥ q ≥ and m ≥ p ≥ and D be any bounded complete n -circular domainwith center at the origin in C n . Then the ( p, q )- th relative Gol’dberg type and ( p, q )- threlative Gol’dberg lower type of f ( z ) with respect to g ( z ) are defined as ∆ ( p,q ) g,D ( f ) = lim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) and ∆ ( p,q ) g,D ( f ) = lim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) , < ρ ( p,q ) g,D ( f ) < + ∞ . Analogously to determine the relative growth of two entire functions of n -complexvariables having same non zero finite p, q ) -th relative Gol’dberg lower order with respectto another entire function of n -complex variables, one may introduce the definition of( p, q ) - th relative Gol’dberg weak type in the following way: Definition 10.
Let f ( z ) and g ( z ) be any two entire functions of n -complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where p, q, m are positive integers suchthat m ≥ q ≥ and m ≥ p ≥ and D be any bounded complete n -circular domainwith center at the origin in C n . Then ( p, q )- th relative Gol’dberg weak type denotedby τ ( p,q ) g,D ( f ) of an entire function f ( z ) with respect to another entire function g ( z ) isdefined as follows: τ ( p,q ) g,D ( f ) = lim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i λ ( p,q ) g ( f ) , < λ ( p,q ) g,D ( f ) < + ∞ . OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 7
Similarly the growth indicator τ ( p,q ) g,D ( f ) of an entire function f ( z ) with respect toanother entire function g ( z ) both of n -complex variables in the following manner : τ ( p,q ) g,D ( f ) = lim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i λ ( p,q ) g ( f ) , < λ ( p,q ) g,D ( f ) < + ∞ . Therefore, for any two entire functions f ( z ) and g ( z ) both of n -complex variables,we note that ρ ( p,q ) g,D ( f ) = λ ( p,q ) g,D ( f ) , ∆ ( p,q ) g,D ( f ) > ⇒ τ ( p,q ) g,D ( f ) = + ∞ and ρ ( p,q ) g,D ( f ) = λ ( p,q ) g,D ( f ) , ∆ ( p,q ) g,D ( f ) > ⇒ τ ( p,q ) g,D ( f ) = + ∞ . Since Gol’dberg has shown that [5]
Gol’dberg type depends on the domain D. Hence all the growth indicators define in Definition 9 and Definition 10 are also dependon D. If f ( z ) and g ( z ) both of n -complex variables have got index-pair ( m,
1) and ( m, l ) , respectively, then the above two definitions reduces to the definition of generalized relativeGol’dberg type ∆ [ l ] g,D ( f ) (resp generalized relative Gol’dberg lower type ∆ [ l ] g,D ( f )) and generalized relative Gol’dberg weak type τ [ l ] g,D ( f ). If the entire functions f ( z ) and g ( z )(both of n -complex variables ) have the same index-pair ( p,
1) where p is any positiveinteger, we get the definitions of relative Gol’dberg type as introduced by Roy [7] (resp. relative Gol’dberg lower type ) and relative Gol’dberg weak type .During the past decades, several authors { cf. [1],[2],[3],[6],[7], [8] } made closedinvestigations on the properties of entire functions of several complex variables usingdifferent growth indicator such as Gol’dberg order, ( p, q ) -th Gol’dberg order etc. In thispaper we wish to measure some properties of entire functions relative to another entirefunction of several complex variables and D will represent a bounded complete n -circulardomain. Actualluy in this paper we wish to study some relative growth properties ofentire functions of n -complex variables (all the entire functions under consideration willbe transcendental unless otherwise stated) using ( p, q ) -th relative Gol’dberg order, ( p, q )-th relative Gol’dberg type and ( p, q )- th relative Gol’dberg weak type. Main Results.
In this section we state the main results of the paper.
Theorem 1.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables and D be a bounded complete n -circular domain with center at origin in C n . Then ( p, q ) -threlative Gol’dberg order ρ ( p,q ) g,D ( f ) and ( p, q ) -th relative Gol’dberg lower order λ ( p,q ) g,D ( f ) of f ( z ) with respect to g ( z ) is independent of the choice of the domain D where p and q are any positive integers . Proof.
Let us consider D and D ne any two bounded complete n -circular domains.Then there exist two real numbers α , β > αD ⊂ D ⊂ βD and therefore M f,αD ( R ) ≤ M f,D ( R ) ≤ M f,βD ( R ) . TANMAY BISWAS
Hence for any bounded complete n -circular domain D,M − g,D ( M f,αD ( R )) ≤ M − g,D ( M f,D ( R )) ≤ M − g,D ( M f,βD ( R )) . (2.1)Now for any θ > D, we get that M f,θD ( R ) = M f,D ( θR ) . Thereforelim R →∞ log [ p ] M − g,D M f,θD ( R )log [ q ] R = lim R →∞ log [ p ] M − g,D M f,D ( θR )log [ q ] R = lim Rθ →∞ log [ p ] M − g,D M f,D ( R )log [ q ] Rθ = lim Rθ →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R + O (1)= lim Rθ →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R .
Hence by (2 .
1) we obtain thatlim R →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R = lim R →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R .
Similarly one can easily verify thatlim R →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R = lim R →∞ log [ p ] M − g,D M f,D ( R )log [ q ] R .
Hence the theorem follows. (cid:3)
Since ρ ( p,q ) g,D ( f ) and λ ( p,q ) g,D ( f ) are independent of the choice of the domain D , andtherefore we write ρ ( p,q ) g ( f ) and λ ( p,q ) g ( f ) instead of ρ ( p,q ) g,D ( f ) and λ ( p,q ) g,D ( f ) respectively. Theorem 2.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be abounded complete n -circular domain with center at origin in C n . Then λ f ( m, q ) ρ g ( m, p ) ≤ λ ( p,q ) g ( f ) ≤ min (cid:26) λ f ( m, q ) λ g ( m, p ) , ρ f ( m, q ) ρ g ( m, p ) (cid:27) ≤ max (cid:26) λ f ( m, q ) λ g ( m, p ) , ρ f ( m, q ) ρ g ( m, p ) (cid:27) ≤ ρ ( p,q ) g ( f ) ≤ ρ f ( m, q ) λ g ( m, p ) . Proof.
From the definitions of ρ ( p,q ) g ( f ) and λ ( p,q ) g ( f ) we get thatlog ρ ( p,q ) g ( f ) = lim R → + ∞ h log [ p +1] M − g,D ( R ) − log [ q +1] M − f,D ( R ) i , (2.2)and log λ ( p,q ) g,D ( f ) = lim R → + ∞ h log [ p +1] M − g,D ( R ) − log [ q +1] M − f,D ( R ) i . (2.3) OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 9
Now from the definitions of ρ f ( m, q ) and λ f ( m, q ) , it follows thatlog ρ f ( m, q ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) i and (2.4)log λ f ( m, q ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) i . (2.5)Similarly, from the definitions of ρ g ( m, p ) and λ g ( m, p ) , we obtain thatlog ρ g ( m, p ) = lim R → + ∞ h log [ m +1] R − log [ p +1] M − g,D ( R ) i and (2.6)log λ g ( m, p ) = lim R → + ∞ h log [ m +1] R − log [ p +1] M − g,D ( R ) i . (2.7)Therefore from (2 . , (2 .
5) and (2 . λ ( p,q ) g ( f ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)i i.e., log λ ( p,q ) g ( f ) ≥ (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log λ ( p,q ) g ( f ) ≥ (cid:0) log λ f ( m, q ) − log ρ g ( m, p ) (cid:1) . (2.8)Similarly, from (2 . , (2 .
4) and (2 . ρ ( p,q ) g ( f ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)i i.e., log ρ ( p,q ) g ( f ) ≤ (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log ρ ( p,q ) g ( f ) ≤ (cid:0) log ρ f ( m, q ) − log λ g ( m, p ) (cid:1) . (2.9)Again, in view of (2 . , (2 . , (2 . , (2 .
6) and (2 . λ ( p,q ) g ( f ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)i i.e., log λ ( p,q ) g ( f ) ≤ min (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) + lim R → + ∞ − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17) , lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) + lim R → + ∞ − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log λ ( p,q ) g ( f ) ≤ min (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17) , lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log λ ( p,q ) g ( f ) ≤ min (cid:8) log λ f ( m, q ) − log λ g ( m, p ) , log ρ f ( m, q ) − log ρ g ( m, p ) (cid:9) . (2.10)Further from (2 . , (2 . , (2 . , (2 .
6) and (2 . ρ ( p,q ) g ( f ) = lim R → + ∞ h log [ m +1] R − log [ q +1] M − f,D ( R ) − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)i i.e., log ρ ( p,q ) g ( f ) ≥ max (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) + lim R → + ∞ − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17) , lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) + lim R → + ∞ − (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log ρ ( p,q ) g ( f ) ≥ max (cid:20) lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17) , lim R → + ∞ (cid:16) log [ m +1] R − log [ q +1] M − f,D ( R ) (cid:17) − lim R → + ∞ (cid:16) log [ m +1] R − log [ p +1] M − g,D ( R ) (cid:17)(cid:21) i.e., log ρ ( p,q ) g ( f ) ≥ max (cid:8) log λ f ( m, q ) − log λ g ( m, p ) , log ρ f ( m, q ) − log ρ g ( m, p ) (cid:9) . (2.11)Thus the theorem follows from (2 . , (2 . , (2 .
10) and (2 . . (cid:3) In view of Theorem 2, one can easily verify the following corollaries:
Corollary 1.
Let f ( z ) be an entire function of n - complex variables with index-pair ( m, q ) and and g ( z ) be any two entire functions of n - complex variables of regular ( m, p ) -th Gol’dberg growth where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Then λ ( p,q ) g ( f ) = λ f ( m, q ) ρ g ( m, p ) and ρ ( p,q ) g ( f ) = ρ f ( m, q ) ρ g ( m, p ) . Moreover, if ρ f ( m, q ) = ρ g ( m, p ) , then ρ ( p,q ) g ( f ) = λ ( q,p ) f ( g ) = 1 . Corollary 2.
Let f and g be any two entire functions of n - complex variables and withregular ( m, q ) -th Gol’dberg growth and regular ( m, p ) -th Gol’dberg growth, respectively, OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 11 where m ≥ q ≥ and m ≥ p ≥ . Also and D be a bounded complete n -circular domainwith center at origin in C n . Then λ ( p,q ) g ( f ) = ρ ( p,q ) g ( f ) = ρ f ( m, q ) ρ g ( m, p ) . Corollary 3.
Let f and g be any two entire functions of n - complex variables and withregular ( m, q ) -th Gol’dberg growth and regular ( m, p ) -th Gol’dberg growth, respectively,where m ≥ q ≥ and m ≥ p ≥ . Also and D be a bounded complete n -circular domainwith center at origin in C n and ρ f ( m, q ) = ρ g ( m, p ) . Then λ ( p,q ) g ( f ) = ρ ( p,q ) g ( f ) = λ ( q,p ) f ( g ) = ρ ( q,p ) f ( g ) = 1 . Corollary 4.
Let f and g be any two entire functions of n - complex variables and withregular ( m, q ) -th Gol’dberg growth and regular ( m, p ) -th Gol’dberg growth, respectively,where m ≥ q ≥ and m ≥ p ≥ . Also and D be a bounded complete n -circular domainwith center at origin in C n . Then ρ ( p,q ) g ( f ) .ρ ( q,p ) f ( g ) = λ ( p,q ) g ( f ) .λ ( q,p ) f ( g ) = 1 . Corollary 5.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . If either f is notof regular ( m, q ) -th Gol’dberg growth or g is not of regular ( m, p ) -th Gol’dberg growth,then λ ( p,q ) g ( f ) .λ ( q,p ) f ( g ) < < ρ ( p,q ) g ( f ) .ρ ( q,p ) f ( g ) . Corollary 6.
Let f ( z ) be an entire function of n - complex variables with index-pair ( m, q ) where m ≥ q ≥ and D be a bounded complete n -circular domain with center atorigin in C n . Then for any entire function g of n - complex variables ( i ) λ ( p,q ) g ( f ) = ∞ when ρ g ( m, p ) = 0 , ( ii ) ρ ( p,q ) g ( f ) = ∞ when λ g ( m, p ) = 0 , ( iii ) λ ( p,q ) g ( f ) = 0 when ρ g ( m, p ) = ∞ and ( iv ) ρ ( p,q ) g ( f ) = 0 when λ g ( m, p ) = ∞ , where m ≥ p ≥ . Corollary 7.
Let g ( z ) be an entire function of n - complex variables with index-pair ( m, p ) where m ≥ p ≥ and D be a bounded complete n -circular domain with center atorigin in C n . Then for any entire function f of n - complex variables ( i ) ρ ( p,q ) g ( f ) = 0 when ρ f ( m, q ) = 0 , ( ii ) λ ( p,q ) g ( f ) = 0 when λ f ( m, q ) = 0 , ( iii ) ρ ( p,q ) g ( f ) = ∞ when ρ f ( m, q ) = ∞ , and ( iv ) λ ( p,q ) g ( f ) = ∞ when λ f ( m, q ) = ∞ , where m ≥ q ≥ . Remark 1.
From the conclusion Theorem 2, one may write ρ ( p,q ) g ( f ) = ρ f ( m,q ) ρ g ( m,p ) and λ ( p,q ) g ( f ) = λ f ( m,q ) λ g ( m,p ) when g ( z ) be an entire function of n - complex variables with regular ( m, p ) -Gol’dberg growth. Similarly ρ ( p,q ) g ( f ) = λ f ( m,q ) λ g ( m,p ) and λ ( p,q ) g ( f ) = ρ f ( m,q ) ρ g ( m,p ) when f ( z ) be an entire function of n - complex variables with regular ( m, q ) - Gol’dberg growth. When f ( z ) and g ( z ) are any two entire functions both of n - complex variables andwith index-pair ( m, q ) and ( n, p ) , respectively, where m ≥ q + 1 ≥ n ≥ p + 1 ≥ , but m = n , the next definition enables us to study their relative order for any boundedcomplete n -circular domain D with center at origin in C n . Definition 11.
Let f ( z ) and g ( z ) be any two entire functions of n - complex vari-ables with index-pair ( m, q ) and ( n, p ) , respectively, where m ≥ q ≥ and n ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . thenthe ( p + m − n, q ) -th relative Gol’dberg order (resp. ( p + m − n, q ) -th relative Gol’dberglower order) of f ( z ) with respect to g ( z ) is defined as ( i ) ρ ( p + m − n,q ) g ( f ) = lim R → + ∞ log [ p + m − n ] M − g,D M f,D ( R )log [ q ] R ( resp. λ ( p + m − n,q ) g ( f ) = lim R → + ∞ log [ p + m − n ] M − g,D M f,D ( R )log [ q ] R ) . If m < n , then the ( p, q + n − m ) -th relative Gol’dberg order (resp. ( p, q + n − m ) -threlative Gol’dberg lower order) of f ( z ) with respect to g ( z ) is defined as ( ii ) ρ ( p,q + n − m ) g ( f ) = lim R → + ∞ log [ p ] M − g,D M f,D ( R )log [ q + n − m ] R ( resp. λ ( p,q + n − m ) g ( f ) = lim R → + ∞ log [ p ] M − g,D M f,D ( R )log [ q + n − m ] R ) . Move to the left.
Theorem 3.
Under the hypothesis of Definition 11, for m > n :( i ) ρ ( p + m − n,q ) g ( f ) = lim R → + ∞ log [ m ] M f,D ( R )log [ q ] R , λ ( p + m − n,q ) g ( f ) = lim R → + ∞ log [ m ] M f,D ( R )log [ q ] R . and for m < n : ( ii ) ρ ( p,q + n − m ) g ( f ) = lim R → + ∞ log [ p ] R log [ n ] M g,D ( R ) , λ ( p,q + n − m ) g ( f ) = lim R → + ∞ log [ p ] R log [ n ] M g,D ( R ) . OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 13
In the next theorem we intend to find out ( p, q )-th relative Gol’dberg order (resp. ( p, q )-th relative Gol’dberg lower order ) of an entire function f ( z ) with respectto another entire function g ( z ) (both f ( z ) and g ( z ) are of n - complex variables ) when( m, q )-th relative Gol’dberg order (resp. ( m, q )-th relative Gol’dberg lower order) of f ( z )and ( m, p )-th relative Gol’dberg order (resp. ( m, p )-th relative Gol’dberg lower order) of g ( z ) with respect to another entire function h ( z ) ( h ( z ) is also of n - complex variables) are given where p, q and m are any positive integers . Theorem 4.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complex vari-ables and D be a bounded complete n -circular domain with center at origin in C n . Alsolet m, p, q are any three positive integers. If ( m, q ) -th relative Gol’dberg order (resp. ( m, q ) -th relative Gol’dberg lower order) of f ( z ) with respect to h ( z ) and ( m, p ) -threlative Gol’dberg order (resp. ( m, p ) -th relative Gol’dberg lower order) of g ( z ) withrespect to h ( z ) are respectively denoted by ρ ( m,q ) h ( f ) (cid:16) resp. λ ( m,q ) h ( f ) (cid:17) and ρ ( m,p ) h ( g ) (cid:16) resp. λ ( m,p ) h ( g ) (cid:17) , then λ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) ≤ λ ( p,q ) g ( f ) ≤ min ( λ ( m,q ) h ( f ) λ ( m,p ) h ( g ) , ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) ) ≤ max ( λ ( m,q ) h ( f ) λ ( m,p ) h ( g ) , ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) ) ≤ ρ ( p,q ) g ( f ) ≤ ρ ( m,q ) h ( f ) λ ( m,p ) h ( g ) . The conclusion of the above theorem can be carried out after applying the sametechnique of Theorem 2 and therefore its proof is omitted.In view of Theorem 4, one can easily verify the following corollaries:
Corollary 8.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) be an entire function with regular relative ( m, q ) -Gol’dberg growth withrespect to entire function h ( z ) and g ( z ) be entire having relative index-pair ( m, p ) withrespect to another entire function h ( z ) where m, p, q are any three positive integers. Then λ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) and ρ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) λ ( m,p ) h ( g ) . In addition, if ρ ( m,q ) h ( f ) = ρ ( m,p ) h ( g ) , then λ ( p,q ) g ( f ) = ρ ( q,p ) f ( g ) = 1 . Corollary 9.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) be an entire function with relative index-pair ( m, q ) with respect to entirefunction h ( z ) and g ( z ) be entire of regular relative ( m, p ) -Gol’dberg growth with respectto another entire function h ( z ) where m, p, q are any three positive integers. Then λ ( p,q ) g ( f ) = λ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) and ρ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) . In addition, if ρ ( m,q ) h ( f ) = ρ ( m,p ) h ( g ) , then ρ ( p,q ) g ( f ) = λ ( q,p ) f ( g ) = 1 . Corollary 10.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be any two entire functions with regular relative ( m, q ) -Gol’dberggrowth and regular relative ( m, p ) -th Gol’dberg growth with respect to entire function h ( z ) respectively where m, p, q are any three positive integers. Then λ ( p,q ) g ( f ) = ρ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) . Corollary 11.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be any two entire functions with regular relative ( m, q ) - Gol’dberggrowth and regular relative ( m, p ) - Gol’dberg growth with respect to entire function h ( z ) respectively where m, p, q are any three positive integers.Then λ ( p,q ) g ( f ) = ρ ( p,q ) g ( f ) = λ ( q,p ) f ( g ) = ρ ( q,p ) f ( g ) = 1 if ρ ( m,q ) h ( f ) = ρ ( m,p ) h ( g ) . Corollary 12.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be any two entire functions with relative index-pairs ( m, q ) and ( m, p ) with respect to entire function h ( z ) respectively where m, p, q are any three positiveintegers and either f ( z ) is not of regular relative ( m, q ) - Gol’dberg growth or g ( z ) isnot of regular relative ( m, p ) - Gol’dberg growth, then ρ ( p,q ) g ( f ) .ρ ( q,p ) f ( g ) ≥ . If f ( z ) and g ( z ) are both of regular relative ( m, q ) - Gol’dberg growth and regular relative ( m, p ) - Gol’dberg growth with respect to entire function h ( z ) respectively, then ρ ( p,q ) g ( f ) .ρ ( q,p ) f ( g ) = 1 . Corollary 13.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be any two entire functions with relative index-pairs ( m, q ) and ( m, p ) with respect to entire function h ( z ) respectively where m, p, q are any three positiveintegers and either f ( z ) is not of regular relative ( m, q ) - Gol’dberg growth or g ( z ) isnot of regular relative ( m, p ) - Gol’dberg growth, then λ ( p,q ) g ( f ) .λ ( q,p ) f ( g ) ≤ . If f ( z ) and g ( z ) are both of regular relative ( m, q ) - Gol’dberg growth and regular relative ( m, p ) -Gol’dberg growth with respect to entire function h ( z ) respectively, then λ ( p,q ) g ( f ) .λ ( q,p ) f ( g ) = 1 . OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 15
Corollary 14.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) be an entire function with relative index-pair ( m, q ) , Then ( i ) λ ( p,q ) g ( f ) = ∞ when ρ ( m,p ) h ( g ) = 0 , ( ii ) ρ ( p,q ) g ( f ) = ∞ when λ ( m,p ) h ( g ) = 0 , ( iii ) λ ( p,q ) g ( f ) = 0 when ρ ( m,p ) h ( g ) = ∞ and ( iv ) ρ ( p,q ) g ( f ) = 0 when λ ( m,p ) h ( g ) = ∞ , where m, p, q are any three positive integers. Corollary 15.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let g ( z ) be an entire function with relative index-pair ( m, p ) , Then ( i ) ρ ( p,q ) g ( f ) = 0 when ρ ( m,q ) h ( f ) = 0 , ( ii ) λ ( p,q ) g ( f ) = 0 when λ ( m,q ) h ( f ) = 0 , ( iii ) ρ ( p,q ) g ( f ) = ∞ when ρ ( m,q ) h ( f ) = ∞ and ( iv ) λ ( p,q ) g ( f ) = ∞ when λ ( m,q ) h ( f ) = ∞ , where m, p, q are any three positive integers. Remark 2.
Under the same conditions of Theorem 4, one may write ρ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) and λ ( p,q ) g ( f ) = λ ( m,q ) h ( f ) λ ( m,p ) h ( g ) when λ ( m,p ) h ( g ) = ρ ( m,p ) h ( g ) . Similarly ρ ( p,q ) g ( f ) = λ ( m,q ) h ( f ) λ ( m,p ) h ( g ) and λ ( p,q ) g ( f ) = ρ ( m,q ) h ( f ) ρ ( m,p ) h ( g ) when λ ( m,q ) h ( f ) = ρ ( m,q ) h ( f ) . Next we prove our theorem based on ( p, q ) -th relative Gol’dberg type and ( p, q ) -threlative Gol’dberg weak type of entire functions of n -complex variables Theorem 5.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be abounded complete n -circular domain with center at origin in C n . Then max (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ≤ ∆ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) . Proof.
From the definitions of ∆ f,D ( m, q ) and ∆ f,D ( m, q ) , we have for all sufficientlylarge values of R that M f,D ( R ) ≤ exp [ m − (cid:20) (∆ f,D ( m, q ) + ε ) h log [ q − R i ρ f ( m,q ) (cid:21) , (2.12) M f,D ( R ) ≥ exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) − ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21) (2.13) and also for a sequence of values of R tending to infinity, we get that M f,D ( R ) ≥ exp [ m − (cid:20) (∆ f,D ( m, q ) − ε ) h log [ q − R i ρ f ( m,q ) (cid:21) , (2.14) M f,D ( R ) ≤ exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) + ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21) . (2.15)Similarly from the definitions of ∆ g,D ( m, p ) and ∆ g,D ( m, p ) , it follows for allsufficiently large values of R that M g,D ( R ) ≤ exp [ m − (cid:20) (∆ g,D ( m, p ) + ε ) h log [ p − R i ρ g ( m,p ) (cid:21) i.e., R ≤ M − g,D (cid:20) exp [ m − (cid:20) (∆ g,D ( m, p ) + ε ) h log [ p − R i ρ g ( m,p ) (cid:21)(cid:21) i.e., M − g,D ( R ) ≥ exp [ p − log [ m − R (∆ g,D ( m, p ) + ε ) ! ρg ( m,p ) and (2.16) M − g,D ( R ) ≤ exp [ p − log [ m − R (cid:0) ∆ g,D ( m, p ) − ε (cid:1) ! ρg ( m,p ) . (2.17)Also for a sequence of values of R tending to infinity, we obtain that M − g,D ( R ) ≤ exp [ p − log [ m − R (∆ g,D ( m, p ) − ε ) ! ρg ( m,p ) and (2.18) M − g,D ( R ) ≥ exp [ p − log [ m − R (cid:0) ∆ g,D ( m, p ) + ε (cid:1) ! ρg ( m,p ) . (2.19)From the definitions of τ f,D ( m, q ) and τ f,D ( m, q ), we have for all sufficientlylarge values of R that M f.D ( R ) ≤ exp [ m − (cid:20) ( τ f,D ( m, q ) + ε ) h log [ q − R i λ f ( m,q ) (cid:21) , (2.20) M f,D ( R ) ≥ exp [ m − (cid:20) ( τ f,D ( m, q ) − ε ) h log [ q − R i λ f ( m,q ) (cid:21) (2.21)and also for a sequence of values of R tending to infinity, we get that M f,D ( R ) ≥ exp [ m − (cid:20) ( τ f.D ( m, q ) − ε ) h log [ q − R i λ f ( m,q ) (cid:21) , (2.22) M f,D ( R ) ≤ exp [ m − (cid:20) ( τ f,D ( m, q ) + ε ) h log [ q − R i λ f ( m,q ) (cid:21) . (2.23) OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 17
Similarly from the definitions of τ g,D ( m, p ) and τ g,D ( m, p ) , it follows for allsufficiently large values of R that M g,D ( R ) ≤ exp [ m − (cid:20) ( τ g,D ( m, p ) + ε ) h log [ p − R i λ g ( m,p ) (cid:21) i.e., R ≤ M − g,D (cid:20) exp [ m − (cid:20) ( τ g,D ( m, p ) + ε ) h log [ p − R i λ g ( m,p ) (cid:21)(cid:21) i.e., M − g,D ( R ) ≥ exp [ p − log [ m − R ( τ g,D ( m, p ) + ε ) ! λg ( m,p ) and (2.24) M − g,D ( R ) ≤ exp [ p − log [ m − R ( τ g,D ( m, p ) − ε ) ! λg ( m,p ) . (2.25)Also for a sequence of values of R tending to infinity, we obtain that M − g,D ( R ) ≤ exp [ p − log [ m − R ( τ g,D ( m, p ) − ε ) ! λg ( m,p ) and (2.26) M − g,D ( R ) ≥ exp [ p − log [ m − R ( τ g,D ( m, p ) + ε ) ! λg ( m,p ) . (2.27)Now from (2 .
14) and in view of (2 . R tendingto infinity that M − g,D M f,D ( R ) ≥ M − g,D (cid:20) exp [ m − (cid:20) (∆ f,D ( m, q ) − ε ) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g,D M f,D ( R ) ≥ exp [ p − log [ m − exp [ m − (cid:20) (∆ f,D ( m, q ) − ε ) h log [ q − R i ρ f ( m,q ) (cid:21) ( τ g,D ( m, p ) + ε ) λg ( m,p ) i.e., log [ p − M − g,D M f,D ( R ) ≥ (cid:20) (∆ f,D ( m, q ) − ε )( τ g,D ( m, p ) + ε ) (cid:21) λg ( m,p ) · h log [ q − R i ρf ( m,q ) λg ( m,p ) . Since in view of Theorem 2, ρ f ( m,q ) λ g ( m,p ) ≥ ρ ( p,q ) g ( f ) and as ε ( >
0) is arbitrary, thereforeit follows from above thatlim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . (2.28)Similarly from (2 .
13) and in view of (2 . R tending to infinity that M − g,D M f,D ( R ) ≥ M − g,D (cid:20) exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) − ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g,D M f,D ( R ) ≥ exp [ p − log [ m − exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) − ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21) ( τ g,D ( m, p ) + ε ) λg ( m,p ) i.e., log [ p − M − g,D M f,D ( R ) ≥ " (cid:0) ∆ f,D ( m, q ) − ε (cid:1) ( τ g,D ( m, p ) + ε ) λg ( m,p ) · h log [ q − R i ρf ( m,q ) λg ( m,p ) . Since in view of Theorem 2, it follows that ρ f ( m,q ) λ g ( m,p ) ≥ ρ ( p,q ) g ( f ) . Also ε ( >
0) isarbitrary, so we get from above thatlim R → + ∞ log [ p − M − g,D M f ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . (2.29)Again in view of (2 . .
12) for all sufficiently large values of R that M − g,D M f,D ( R ) ≤ M − g,D (cid:20) exp [ m − (cid:20) (∆ f,D ( m, q ) + ε ) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g,D M f,D ( R ) ≤ exp [ p − log [ m − exp [ m − (cid:20) (∆ f,D ( m, q ) + ε ) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:0) ∆ g,D ( m, p ) − ε (cid:1) ρg ( m,p ) OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 19 i.e., log [ p − M − g,D M f,D ( R ) ≤ " (∆ f,D ( m, q ) + ε ) (cid:0) ∆ g,D ( m, p ) − ε (cid:1) ρg ( m,p ) · h log [ q − R i ρf ( m,q ) ρg ( m,p ) . (2.30)As in view of Theorem 2, it follows that ρ f ( m,q ) ρ g ( m,p ) ≤ ρ ( p,q ) g ( f ) Since ε ( >
0) is arbi-trary, we get from (2 .
30) thatlim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) . (2.31)Thus the theorem follows from (2 . .
29) and (2 . (cid:3) The conclusion of the following corollary can be carried out from (2 .
17) and (2 . .
20) and (2 .
25) respectively after applying the same technique of Theorem 5 and withthe help of Theorem 2. Therefore its proof is omitted.
Corollary 16.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Then ∆ ( p,q ) g,D ( f ) ≤ min ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ) . Similarly in the line of Theorem 5 and with the help of Theorem 2, one may easilycarried out the following theorem from pairwise inequalities numbers (2 .
21) and (2 .
24) ;(2 .
18) and (2 . .
17) and (2 .
23) respectively and therefore its proofs is omitted:
Theorem 6.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be abounded complete n -circular domain with center at origin in C n . Then (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ≤ τ ( p,q ) g,D ( f ) ≤ min ((cid:20) τ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) τ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ) . Corollary 17.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Then τ ( p,q ) g,D ( f ) ≥ max (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . With the help of Theorem 2, the conclusion of the above corollary can be carryout from (2 . , (2 .
16) and (2 . , (2 .
24) respectively after applying the same techniqueof Theorem 5 and therefore its proof is omitted.
Theorem 7.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be abounded complete n -circular domain with center at origin in C n . Then (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ≤ ∆ ( p,q ) g,D ( f ) ≤ min (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) . Proof.
From (2 .
13) and in view of (2 . R that M − g,D M f,D ( R ) ≥ M − g (cid:20) exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) − ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g M f ( R ) ≥ exp [ p − log [ m − exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) − ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21) ( τ g,D ( m, p ) + ε ) λg ( m,p ) i.e., log [ p − M − g,D M f,D ( R ) ≥ " (cid:0) ∆ f,D ( m, q ) − ε (cid:1) ( τ g,D ( m, p ) + ε ) λg ( m,p ) · h log [ q − R i ρf ( m,q ) λg ( m,p ) . Now in view of Theorem 2, it follows that ρ f ( m,q ) λ g ( m,p ) ≥ ρ ( p,q ) g ( f ) . Since ε ( >
0) isarbitrary, we get from above thatlim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≥ (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . (2.32)Further in view of (2 . , we get from (2 .
12) for a sequence of values of R tendingto infinity that M − g,D M f,D ( R ) ≤ M − g,D (cid:20) exp [ m − (cid:20) (∆ f,D ( m, q ) + ε ) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g,D M f,D ( R ) ≤ exp [ p − log [ m − exp [ m − (cid:20) (∆ f,D ( m, q ) + ε ) h log [ q − R i ρ f ( m,q ) (cid:21) (∆ g,D ( m, p ) − ε ) ρg ( m,p ) i.e., log [ p − M − g,D M f,D ( R ) ≤ (cid:20) (∆ f,D ( m, q ) + ε )(∆ g,D ( m, p ) − ε ) (cid:21) ρg ( m,p ) · h log [ q − R i ρf ( m,q ) ρg ( m,p ) . (2.33) OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 21
Again as in view of Theorem 2, ρ f ( m,q ) ρ g ( m,p ) ≤ ρ ( p,q ) g ( f ) and ε ( >
0) is arbitrary, thereforewe get from (2 .
33) thatlim R → + ∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) . (2.34)Likewise from (2 .
15) and in view of (2 . R tending to infinity that M − g,D M f,D ( R ) ≤ M − g,D (cid:20) exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) + ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:21) i.e., M − g,D M f,D ( R ) ≤ exp [ p − log [ m − exp [ m − (cid:20)(cid:0) ∆ f,D ( m, q ) + ε (cid:1) h log [ q − R i ρ f ( m,q ) (cid:21)(cid:0) ∆ g,D ( m, p ) − ε (cid:1) ρg ( m,p ) i.e., log [ p − M − g,D M f,D ( R ) ≤ " (cid:0) ∆ f,D ( m, q ) + ε (cid:1)(cid:0) ∆ g,D ( m, p ) − ε (cid:1) ρg ( m,p ) · h log [ q − R i ρf ( m,q ) ρg ( m,p ) . (2.35)Analogously, we get from (2 .
35) thatlim r →∞ log [ p − M − g,D M f,D ( R ) h log [ q − R i ρ ( p,q ) g ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) i.e., ∆ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (2.36)since in view of Theorem 2, ρ f ( m,q ) ρ g ( m,p ) ≤ ρ ( p,q ) g ( f ) and ε ( >
0) is arbitrary.Thus the theorem follows from (2 . .
34) and (2 . (cid:3) Corollary 18.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Then ∆ ( p,q ) g,D ( f ) ≤ min ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) σ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) τ f,D ( m, q ) σ g,D ( m, p ) (cid:21) ρg ( m,p ) ) . The conclusion of the above corollary can be carried out from pairwise inequalitiesno (2 .
17) and (2 .
23) ; (2 .
18) and (2 .
20) ; (2 .
23) and (2 . .
20) and (2 .
26) respectivelyafter applying the same technique of Theorem 7 and with the help of Theorem 2. There-fore its proof is omitted.Similarly in the line of Theorem 5 and with the help of Theorem 2, one may easilycarried out the following theorem from pairwise inequalities no (2 .
22) and (2 .
24) ; (2 . . .
17) and (2 .
20) respectively and therefore its proofs is omitted:
Theorem 8.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be abounded complete n -circular domain with center at origin in C n . Then max ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ) ≤ τ ( p,q ) g,D ( f ) ≤ (cid:20) τ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) . Corollary 19.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Then τ ( p,q ) g,D ( f ) ≥ max (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) ∆ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . The conclusion of the above corollary can be carried out from pairwise inequalitiesno (2 .
14) and (2 .
16) ; (2 .
13) and (2 .
19) ; (2 .
14) and (2 . .
13) and (2 .
27) respectivelyafter applying the same technique of Theorem 7 and with the help of Theorem 2. There-fore its proof is omitted.Now we state the following theorems without their proofs as because they can bederived easily using the same technique or with some easy reasoning with the help ofRemark 1 and therefore left to the readers.
Theorem 9.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variables withindex-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D bea bounded complete n -circular domain with center at origin in C n . Also let g ( z ) is ofregular ( m, p ) -Gol’dberg growth. Then (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ ∆ ( p,q ) g,D ( f ) ≤ min (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ max (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ ∆ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 23 and (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ≤ τ ( p,q ) g,D ( f ) ≤ min ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ) ≤ max ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ) ≤ τ ( p,q ) g,D ( f ) ≤ (cid:20) τ f ( m, q ) τ g ( m, p ) (cid:21) λg ( m,p ) . Theorem 10.
Let f ( z ) and g ( z ) be any two entire functions of n - complex variableswith index-pair ( m, q ) and ( m, p ) , respectively, where m ≥ q ≥ and m ≥ p ≥ and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) is ofregular ( m, q ) -Gol’dberg growth. Then (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ τ ( p,q ) g,D ( f ) ≤ min (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ max (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) , (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) ≤ τ ( p,q ) g,D ( f ) ≤ (cid:20) ∆ f,D ( m, q )∆ g,D ( m, p ) (cid:21) ρg ( m,p ) and (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ≤ ∆ ( p,q ) g,D ( f ) ≤ min ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ) ≤ max ((cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) , (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) ) ≤ ∆ ( p,q ) g,D ( f ) ≤ (cid:20) τ f,D ( m, q ) τ g,D ( m, p ) (cid:21) λg ( m,p ) . In the next theorems we intend to find out ( p, q )-th relative Gol’dberg type (resp. ( p, q )-th relative Gol’dberg lower type, ( p, q )-th relative Gol’dberg weak type) ofan entire function f ( z ) with respect to another entire function g ( z ) (both f ( z ) and g ( z ) are of n - complex variables ) when ( m, q )-th relative Gol’dberg type (resp. ( m, q )-th relative Gol’dberg lower type, ( m, q )-th relative Gol’dberg weak type) of f ( z ) and( m, p )-th relative Gol’dberg type (resp. ( m, p )-th relative Gol’dberg lower type, ( m, p )-threlative Gol’dberg weak type) of g ( z ) with respect to another entire function h ( z ) ( h ( z )is also of n - complex variables ) are given where m ≥ p ≥ m ≥ q ≥ . Basically westate the theorems without their proofs as those can easily be carried out after applyingthe same technique our previous discussion and with the help of Theorem 4 and Remark2.
Theorem 11.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be entire functions with relative index-pairs ( m, q ) and ( m, p ) with respect to h ( z ) respectively where p, q, m are all positive integers . If λ ( m,p ) h ( g ) = ρ ( m,p ) h ( g ) , then σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ min σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) ≤ max σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) and τ ( m,q ) h,,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ min τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ max τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) . Theorem 12.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be entire functions with relative index-pairs ( m, q ) and ( m, p ) withrespect to h ( z ) respectively where p, q, m are all positive integers . If f ( z ) is of regularrelative ( m, q ) - Gol’dberg growth with respect to entire function h ( z ) , then σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ min σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) ≤ max σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) σ ( m,p ) h,D, ( g ) ρ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) and τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ min τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ max τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) . Theorem 13.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . OME RESULTS RELATING TO ( p, q )-TH RELATIVE GOL’DBERG ORDER..... 25
Also let f ( z ) and g ( z ) be entire functions with relative index-pairs ( m, q ) and ( m, p ) with respect to h ( z ) respectively where p, q, m are all positive integers . Then max σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ min τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) and σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ σ ( p,q ) g,D ( f ) ≤ min (cid:20) σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) , (cid:20) σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) , (cid:20) τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) , (cid:20) τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) , (cid:20) τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) , (cid:20) τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) . Theorem 14.
Let f ( z ) , g ( z ) and h ( z ) be any three entire functions of n - complexvariables and D be a bounded complete n -circular domain with center at origin in C n . Also let f ( z ) and g ( z ) be entire functions with relative index-pairs ( m, q ) and ( m, p ) with respect to h ( z ) respectively where p, q, m are all positive integers . Then max (cid:20) τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) , (cid:20) τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) , (cid:20) σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) , (cid:20) σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) (cid:21) ρ ( m,p ) h ( g ) , (cid:20) σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) , (cid:20) σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) (cid:21) λ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) and max σ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) , σ ( m,q ) h,D ( f ) τ ( m,p ) h,D ( g ) λ ( m,p ) h ( g ) ≤ τ ( p,q ) g,D ( f ) ≤ min τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) , τ ( m,q ) h,D ( f ) σ ( m,p ) h,D ( g ) ρ ( m,p ) h ( g ) . References [1] S. K. Datta and A. R. Maji : Study of Growth propertis on the basis of generalised Gol’dbergorder of composite entire functions of several complex variables,International J. of Math.Sci.&Engg.Appls.(IJMSEA), Vol. 5 No.V (2011). pp.297-311.[2] S.K. Datta and A.R. Maji : Some study of the comparative growth rates on the basis of generalisedrelative Gol’dberg order of composite entire functions of several complex variables, International J.ofMath. Sci. & Engg. Appls.(IJMSEA), Vol. 5, No. V ( 2011), pp. 335 - 344.[3] S.K. Datta and A.R. Maji : Some study of the comparative growth properties on the basis of relativeGol’dberg order of composite entire functions of several complex variables, Int. J. Contemp. Math.Sciences, Vol. 6, No. 42 (2011), pp. 2075 - 2082.[4] B. A. Fuks : Introduction to the heory of analytic functions of several complex variables, AmericanMathematical Soci., Providence, R. I., 1963.[5] A. A. Gol’dberg : Elementary remarks on the formulas defining order and type of functions of severalvariables, Akad, Nank, Armjan S. S. R. Dokl, Vol. 29 (1959), pp. 145-151 (Russian).[6] B. C. Mondal and C. Roy : Relative gol’dberg order of an entire function of several variables, BullCal. Math. Soc., Vol. 102, No. 4(2010), pp. 371-380.[7] C. Roy: Some properties of entire functions in one and several complex vaiables, Ph.D. Thesis (2010), University of Calcutta.[8] B. Prajapati and A. Rastogi: Some results o p th gol’dberg relative order, International Journal ofApplied Mathematics and Statistical Sciences, Vol. 5, No. 2 (2016), pp. 147-154. T. Biswas : Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.- Krishnagar, Dist-Nadia,PIN- 741101, West Bengal, India
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