Gen. Math. Notes, Vol. 2, No. 1, January 2011, pp. 134-142
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Some Growth Properties of Entire Functions
Represented by Vector Valued Dirichlet
Series in Two Variables
G. S. Srivastava and Archna Sharma
Department of Mathematics,
Indian Institute of Technology Roorkee,
Roorkee- 247667, India.
E-mail: girssfma@iitr.ernet.in.
gs91490@gmail.com.
(Received 15.11.2010, Accepted 20.12.2010)
Abstract
In the present paper, we study the entire functions represented by vector
valued Dirichlet series of several complex variables. The characterizations
of their order and type have been obtained. For the sake of simplicity,
we have considered the functions of two variables only.
Keywords: Entire function, Dirichlet Series, Order, Type.
AMS Subject Classification: 30B50, 30D15, 32A15.
135
1
Some growth properties of entire functions
Introduction
Consider
f ( s1 , s2 ) =
∞
∑a
m , n =1
mn
e( s1λm + s2 µn ) , ( s j = σ j + it j , j = 1, 2)
(1)
where am,n ' s belong to the Banach space ( E ,|| . ||);0 < λ1 < ... < λm → ∞ as m → ∞ ,
0 < µ1 < ... < µn → ∞ as n → ∞,
and
lim sup
m + n →∞
log(m + n)
= D < +∞ .
λm + µ n
(2)
Such a series is called a vector valued Dirichlet series in two complex variables.
The concepts of order and type of an entire function (also for analytic function)
represented by vector valued Dirichlet series of one complex variable were first
introduced in 1983 by O.P.Juneja and B.L.Srivastava.They also obtained the
coefficient characterizations of order and type. In this paper we have extended the
results to the entire functions represented by vector valued Dirichlet series of
several complex variables. For simplicity, we consider here functions of two
variables only, though these results can easily be extended to functions of several
complex variables.
Let f ( s1 , s2 ) defined by (1) represent an entire function. We define
M (σ 1 , σ 2 ) = sup{|| f (σ 1 + it1 , σ 2 + it2 ) ||; − ∞ < t j < ∞, j = 1, 2}
to be the maximum modulus of f ( s1 , s2 ).Then M (σ 1 , σ 2 ) → ∞ as σ 1 , σ 2 → ∞. We
define the order ρ (0 ≤ ρ ≤ ∞) of f ( s1 , s2 ) as follows.
Class A: An entire function f ( s1 , s2 ) defined by vector valued Dirichlet series of
finite order belongs to class A, if there exist positive constants K1 , γ 1 , K 2 and γ 2
such that
(i) For any fixed value of σ 2 > 0 , there exists a number σ (1) = σ (1) ( K1 , γ 1 , σ 2 ) such
that
M (σ 1 , σ 2 ) < exp{K1 exp(σ 1γ 1 )} for σ 1 ≥ σ (1) .
(ii) For any fixed value of σ 1 > 0 , there exists a number
σ (2) = σ (2) ( K 2 , γ 2 , σ 1 ) such that
M (σ 1 , σ 2 ) < exp{K 2 exp(σ 2γ 2 )} for σ 2 ≥ σ (2) .
Therefore, there exists a number σ = σ ( K1 , K 2 , γ 1 , γ 2 ) such that
M (σ 1 , σ 2 ) < exp{K1 exp(σ 1γ 1 ) + K 2 exp(σ 2γ 2 )} for σ 1 , σ 2 ≥ σ .
G. S. Srivastava and Archna Sharma
136
Definition 1.1 An entire function f ( s1 , s2 ) defined by the vector valued Dirichlet
series (1.1) has finite orders ρ1 and ρ 2 with respect to s1 and s2 respectively if
(i) For any arbitrarily small ε > 0 , and any σ 2 > 0 ,there exists a number σ (1) =
σ (1) (ε , σ 2 ) such that
M (σ 1 , σ 2 ) < exp[exp{σ 1 ( ρ1 + ε )}]
for σ 1 ≥ σ (1) .
In addition, there exists at least one value of σ 2 , say σ 2 0 (ε ) and correspondingly
arbitrary large values of σ 1 :{σ 1i } such that
M (σ 1i , σ 2 0 (ε )) > exp exp{σ 1i ( ρ1 − ε )} .
Hence, (i) is equivalent to

log log M (σ 1 , σ 2 ) 
lim sup  lim sup
 = ρ1 .
σ 2 →∞
σ1
σ1 →∞

(ii) For any arbitrarily small ε > 0 , and any σ 1 > 0 ,there exists a number
σ ( 2) = σ ( 2) (ε , σ 1 )
such that
M (σ 1 , σ 2 ) < exp[exp{σ 2 ( ρ 2 + ε )}]
for σ 2 ≥ σ (2) .
In addition, there exists at least one value of σ 1 , say σ 10 (ε ) and correspondingly
arbitrarily large values of σ 2 :{σ 2 j } such that
M (σ 10 (ε ), σ 2 j ) > exp exp{σ 2 j ( ρ 2 − ε )} .
So, (ii) is equivalent to

log log M (σ 1 , σ 2 ) 
lim sup  lim sup
 = ρ2 .
σ1 →∞
σ2
σ 2 →∞

Definition 1.2 An entire function f ( s1 , s2 ) defined by vector valued Dirichlet
series has a finite order ( ρ1 , ρ 2 ) if
(i) f ( s1 , s2 ) ∈ A
(ii) f ( s1 , s2 ) has finite orders ρ1 andρ 2 with respect to s1 and s2 respectively as
above.
(iii) For ε > 0 , there exists a number σ = σ (ε ) such that
M (σ 1 , σ 2 ) < exp {exp σ 1 ( ρ1 + ε ) + exp σ 2 ( ρ 2 + ε )}
Thus, we define the order of vector valued Dirichlet series as
for σ 1 , σ 2 ≥ σ .
137
Some growth properties of entire functions
lim sup
σ1 ,σ 2 →∞
log log M (σ 1 , σ 2 )
=ρ
log(eσ1 + eσ 2 )
.
Similarly, if 0 < ρ < ∞ , then the type T (0 ≤ T ≤ ∞) of f ( s1 , s2 ) is defined as:
lim sup
σ1 ,σ 2 →∞
log M (σ 1 , σ 2 )
=T
(e ρσ1 + e ρσ 2 )
.
2. Basic Results.
We now prove
Theorem 2.1 The necessary and sufficient condition for the series (1) satisfying
the condition (2) to be entire is that
log(|| am ,n ||)
lim sup
= −∞ .
(3)
m + n →∞
λm + µn
For proving our result, we need the following result:
Lemma 2.2. The following conditions are equivalent:
log(m + n)
= D < +∞
m + n →∞
λm + µn
log m
log n
= D1 < +∞ , lim sup
= D2 < +∞ ,
(ii) lim sup
(i)
lim sup
m →∞
λm
µn
n →∞
(iii)There exists α , 0 < α < ∞ , such that the series
∞
∑ exp[−α (λ
m,n=0
m
+ µn )] converges.
For the proof of this Lemma we refer to [2].
Proof of Theorem 2.1 We suppose that (1) defines an entire function.Then it
converges absolutely for all ( s1 , s2 ) .Now take the points with coordinates
∞
( p, p ) with p > 0 .Then it follows that
∑ || a
m,n=0
Therefore
and
m,n
||exp(λm + µn ) p < ∞
|| am, n || exp(λm + µn ) p ≤ M ( p, p )
lim sup
m + n →∞
log || am ,n ||
λm + µn
< − p.
Since p > 0 is arbitrary, the necessity part of the theorem is proved.
G. S. Srivastava and Archna Sharma
138
Conversely, let the given condition (3) be satisfied. It suffices to prove
that (1) converges absolutely for all ( s1 , s2 ) .Let us consider ( s1 , s2 ) ∈ C 2 , σ > 0
such that Re s1 < σ , Re s2 < σ . Then by (3) and for some δ > 0 ,
log || am ,n ||
λm + µn
and so
< −σ − δ for m + n ≥ N 0 (δ )
|| am, n || exp{(λm + µ n )σ } < exp(−δ (λm + µ n )) .
∞
But the double series
∑ exp[−δ (λ
m
m,n=0
(4)
+ µ n )] being convergent in view of Lemma
2.2, it follows from (4) that the series (1) converges absolutely for all ( s1 , s2 ) ,
given by Re s1 < σ , Re s2 < σ . Now, when the series (1) converges at ( s1 , s2 ) it
also converges at ( s1' , s2' ) where Re s1' < Re s1 , Re s2' < Re s2 , hence the result
follows.
Next we prove
Theorem 2.3 If
f ( s1 , s2 ) =
∞
∑a
m , n =1
mn
e( s1λm + s2 µn )
is an entire function of order ( ρ1 , ρ 2 ), (0 < ρ1 , ρ 2 < ∞), then
 log log M (σ 1 , σ 2 ) 
lim sup 
 =1
σ1 ,σ 2 →∞
 σ 1 ρ1 + σ 2 ρ 2 
.
Proof. The proof of this theorem follows on the lines of the proof of Theorem1 in
[2].
3 Main Results
Theorem 3.1 If f ( s1 , s2 ) is an entire function of order ρ (0 < ρ < ∞), then
log(λmλm µ nµn )
ρ = lim sup
m , n →∞
log || amn ||−1
Proof. Let
.
139
Some growth properties of entire functions
µ = lim sup
m , n →∞
log(λmλm µnµn )
log || amn ||−1
First we show that ρ ≥ µ . Let us assume that µ > 0, for otherwise the result is
trivially true.Then for a given ε > 0 , we have two sequences {λM p } and
{µ Nq }with M p → ∞ as p → ∞ and N q → ∞ as q → ∞ such that
log || amn || > − ( µ − ε )−1 (λm log λm + µ n log µ n )
for m = M p and n = N q .
Since the inequality:
M (σ 1 , σ 2 ) ≥ || amn || exp(σ 1λm + σ 2 µn ) holds for all σ 1 , σ 2 and m, n ,it
follows that for all σ 1 , and σ 2 and m = M p and n = N q
log M (σ 1 , σ 2 ) > λm {σ 1 − ( µ − ε ) −1 log λm } + µn {σ 2 − ( µ − ε ) −1 log µn } .
Taking
σ 1, p = ( µ − ε )−1 log(eλM ) and σ 2,q = ( µ − ε ) −1 log(eµ N )
p
q
in the above we find that
log M (σ 1 , σ 2 ) >
exp(σ 1 ( µ − ε )) + exp(σ 2 ( µ − ε ))
e( µ − ε )
for σ 1 = σ 1, p and σ 2 = σ 2,q . Proceeding to limits as σ 1 , σ 2 → ∞, we get ρ ≥ µ .
Further, to prove the converse part, we suppose that µ < ∞ , for otherwise the
result is obviously true, so that
|| amn ||< λm −λm ( µ +ε ) .µn − µn ( µ +ε ) ,
m > m0 and n > n0 .
(5)
Now,
n0
m0
∞
∞
∞
∞
 m0 n0

M (σ 1 , σ 2 ) ≤  ∑∑ + ∑ ∑ + ∑ ∑ + ∑ ∑  || amn || exp(σ 1λm + σ 2 µ n )
 m =1 n =1 m = m0 +1 n =1 n = n0 +1 m =1 m = m0 +1 n = n0 +1 
= ∑ +∑ +∑ + ∑ .
(6)
1
2
3
4
Now we estimate the four parts of (6). Clearly
∑ = O( exp(σ 1λm0 + σ 2 µn0 )).
1
In view of (5), we get
G. S. Srivastava and Archna Sharma
∑ ≤ ∑ ∑ exp{(σ λ
1 m
m > m0 n > n0
4
≤
∑ ∑ exp{σ λ
1 m
m > m0 n > n0
140
+ σ 2 µ n ) + log(λm −λm
( µ +ε )
.µ n − µn
( µ +ε )
)}
+ σ 2 µ n − ( µ + ε ) −1 (λm log λm + µn log µn )}
≤ max[exp(σ 1λm + σ 2 µ n − ( µ + 2ε ) −1 (λm log λm + µn log µ n )].
 λm log λm + µ n log µn 
.
−1
m > m0 n > n0
 ε ( µ + ε )( µ + 2ε ) 
The series on the right hand side of the above inequality is convergent and the
maximum of the expression
exp(σ 1λm + σ 2 µ n − ( µ + 2ε )−1 (λm log λm + µn log µ n )
∑ ∑ exp  −
is attained at λm = e −1 exp(σ 1 ( µ + 2ε )) and µn = e −1 exp(σ 2 ( µ + 2ε )) , we find that
∑
≤ A exp{e −1 ( µ + 2ε )−1 (eσ1 ( µ + 2ε ) + eσ 2 ( µ + 2ε ) )}
4
where A is an absolute constant. Further, to estimate
∑ , µ being finite, we have
2
for all values of m and n
log(λmλm µnµn )
≤ς,
log || amn ||−1
Therefore,
∑≤ ∑
2
n0
∑ exp(σ λ
1 m
m > m0 n =1
+ σ 2 µ n − ς −1λm log λm − ς −1µ n log µn )
= O(exp(σ 2 µn0 )) ∑ exp(σ 1λm − ς −1λm log λm )
m > m0
≤ O(exp(σ 2 µ n0 ) max[exp(σ 1λm − (ς + ε ) −1 λm log λm ). ∑ (−ες −1 (ς + ε ) −1 λm log λm )]
m > m0
= O(exp(σ 2 µn0 ) max[exp(σ 1λm − (ς + ε ) −1 λm log λm )].
But the max .exp(σ 1λm − (ς + ε )−1 λm log λm ) is attained at
λm = e −1 exp(σ 1 (ς + ε )).
Hence,
∑
≤ O(exp(σ 2 µn0 ) exp{e −1 (ς + ε ) −1 exp(σ 1 (ς + ε ))} .
2
Similarly
∑
3
≤ O(exp(σ 1λm0 ) exp{e −1 (ς + ε ) −1 exp(σ 2 (ς + ε ))} .
141
Some growth properties of entire functions
Substituting these estimates of
∑ , (1 ≤ i ≤ 4), in (6), we have
i
log log M (σ 1 , σ 2 ) ≤ log{exp(σ 1 ( µ + 2ε ) + exp(σ 2 ( µ + 2ε )} + O(1) ,
which gives ρ ≤ µ on proceeding to limits as σ 1 , σ 2 → ∞. Hence the theorem
follows.
Theorem 3.2. Let f ( s1 , s2 ) be an entire function of order ρ (0 < ρ < ∞) and type
T (0 ≤ T ≤ ∞). Then
lim sup(λmλm µnµn . || amn ||ρ )1 ( λm + µn ) = eρT .
m , n →∞
Proof. We shall only sketch the proof, since it follows on the lines of proof of
Theorem1.
For a given ε > 0 we get sequences m = M p and n = N q , such that


α −ε 
α −ε 
−1
log M (σ 1 , σ 2 ) > λm  σ 1 + ρ −1 log
 + µn  σ 2 + ρ log

λm 
µn 


(7)
where α is assumed to be positive and is given by
α = lim sup(λmλm µ nµn . || amn ||ρ )1 ( λm + µn ) .
m , n →∞
Choosing in (7) the sequences of the values of σ 1 and σ 2 given by
 eλm p 
 eµnq 
−1
 , σ 2 = ρ log 

α −ε 
α −ε 
it follows that eρT ≥ α . This assertion is trivially true for the case when α = 0.
σ 1 = ρ −1 log 
The converse part follows by using the following estimations of
∑ ' s in (6):
i
∑ ≤ O( exp(σ λ
1 m0
+ σ 2 µ n0 )),
1
 ς + ε ρσ 2 
) exp 
.e  ,
2
 eρ

ς +ε

∑3 ≤ O(exp(σ 2 µn0 ) exp  eρ .e ρσ1  ,


∑ ≤ O(exp(σ λ
1 m0
 α + 2ε ρσ1

.(e + e ρσ 2 ) .
eρ
4

where A1 is an absolute constant. Hence
∑ ≤ A exp 
1
M (σ 1 , σ 2 ) ≤ exp(e ρσ1 + e ρσ 2 ) + O(1) ,
or log M (σ 1 , σ 2 ) ≤ (e ρσ1 + e ρσ 2 ) + o(1)
which gives eρT ≤ α on proceeding to limits as σ 1 , σ 2 → ∞. This proves the result.
G. S. Srivastava and Archna Sharma
References
[1] B.L.Srivastava, A study of spaces of certain classes of Vector Valued
Dirichlet Series, Thesis, I.I.T Kanpur, (1983).
[2] S.Daoud, Entire functions represented by Dirichlet series of two complex
variables, Portugaliae Math., 43(4)(1985-1986), 407-415.
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