On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet Series

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 29-34
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On Spaces of Entire Functions Having Slow Growth Represented
By Dirichlet Series
Arvind Kumar
Department of Mathematics,IMA Dehradun.
Abstract: In this paper spaces of entire function represented by Dirichlet Series have been considered. A
norm has been introduced and a metric has been defined. Properties of this space and a characterization of
continuous linear functionals have been established.
1. Let,
(1.1) f(s) = n . e s.λ
n .
where 0 < λ1 < λ2 < ……… < λn ..……, λn → ∞ as n → ∞, s = σ + it (σ, t being reals) and {an}1
∞
any sequence
of complex numbers, be a Dirichlet Series. Further, Let -:
(1.2) n/λn = D < ∞ .
(1.3) (λn+1 – λn).
and
(1.4) ӏanӏ-1
λn
Then the series in (1.1) represents an entire function f(s). We donate by X the set of all entire functions f(s)
having representation (1.1) and satisfying the conditions (1.2)-(1.4). By giving different topologies on the set X,
Kamthan[4] and Hussain and Kamthan [2] have studied various topological properties of these spaces. Hence
we define, for any non- decreasing sequence {ri} of positive numbers, ri → ∞,
(1.5) ӏӏ f ӏӏ ri = Σ ӏanӏ er
i
λ
n, i=1, 2, 3 .......... where f ϵ X. Then from (1.4), ӏӏ f ӏӏri exists for each i and is a form
of X. Further, ӏӏ f ӏӏ ri ≤ ӏӏ f ӏӏ ri+1. With these countable numbers of norms, a metric d is defined on X as :
(1.6) d (f, g) = i ӏӏ f-g ӏӏ ri
1 + ӏӏ f-g ӏӏ ri
Further, following functions are defined for each f ϵ X, namely
(1.7) p (f) = ӏ an ӏ 1/λ
n.
(1.8) ӏӏ f ӏӏ ri = ӏ an ӏ 1/λ
n.
Then p (f) and ӏӏ f ӏӏi are para-norms on X. Let
(1.9) s (f, g) = i ӏӏ f-g ӏӏ ri , f, g ϵ X.
1+ӏӏ f-g ӏӏ ri
It was shown [2, Lemma 1] that the three topologies induced by d, s and p on X are equivalent. Many other
properties of these spaces were also obtained (see [2], pp. 206-209).
For the space of entire functions of finite Ritt order [6] and type, yet another norm ӏӏ f ӏӏ q and hence metric λ
was introduced and the properties of this space Xλ were studied.

On Spaces Of Entire Functions Having Slow Growth Represented By Dirichlet Series
Let, for f ϵ X,
M (σ, t) M (σ) = sup ӏ fμ (σ + it) ӏ
-∞ < t <∞
Then M (σ) is called the maximum modulus of f(s). The Ritt order of f(s) is defined as
(1.10) = ρ, 0 ≤ ρ ≤ ∞.
σ
For f < ∞, the entire function f is said to be of finite order. A function f(σ) is said to be proximate order [3] if
(1.11) ρ (σ) → ρ as σ → ∞, 0 < ρ < ∞,
(1.12) σ ρ’ (σ) → 0 as σ → ∞.
For f ϵ X, define
(1.13) ≤ A < ∞.
Then it was proved [3] that (1.13) holds if and only if
(1.14) n) ӏanӏ1/λn
< (A.e ρ)1/ρ
, where (t) is the unique solution of the equation t = exp
[ ].
(Apparently the inequality (4.1) and the definition of (t) contain some misprints in [2, pp.209-210]).
For each f ϵ X, define
ӏӏ f ӏӏ q = n n) ,
[(A+1/q)e]1/ρ
Where q = 1, 2, 3............ . For q1 ≤ q2, ӏӏ f ӏӏq1 ≤ ӏӏ f ӏӏq2. It was proved that ӏӏ f ӏӏq, q = 1, 2, 3............. Induces on
X a unique topology such that X becomes a convex topological vector space, where this topology is given by the
metric λ,
this space was donated by Xλ. Various properties of this space were studied [2, pp. 209-216].
It is evident that if ρ = 0, then the definition of the norm ӏӏ f ӏӏ q and proximate order ρ (σ) is not possible. It is
the aim of this paper to give a metric on the space of entire functions of zero order thereby studying some
properties of this space.
2. For an entire function f(s) represented by (1.1), for which ρ defined by (1.10) is equal to zero, we define
following Rahman [5].
(2.1) = ρ*, 1 ≤ ρ* ≤ ∞.
log σ
Then ρ* is said to be the logarithmic order of f(s). For 1 < ρ* < ∞, we define the logarithmic proximate order [1]
ρ*(σ) as a continuous piecewise differentiable function for σ > σ0 such that
(2.2) ρ*(σ) → ρ* as σ → ∞,
(2.3) σ.log σ. ρ*(σ) → 0 as σ → ∞.
Then the logarithmic type T* of {f} with respect to proximate order ρ*( σ) is defined as [7]:
(2.4) = T*, 0 < T* < ∞.
σρ*(σ)

It was proved by one of the authors [7] that f(s) is of logarithmic order ρ*,
1 < ρ* < ∞ and logarithmic type T*, 0 < T* < ∞ if and only if
(2.5) n) = ρ* (ρ*T*) 1/(ρ*-1)
,
log ӏanӏ-1
ρ*-1
where (t) is the unique solution of the equation t = σρ*(σ)-1
. We now denote by X the set of all entire functions
f(s) given by (1.1), satisfying (1.2) to (1.4), for which
(2.6) ≤ T* < ∞, 1 < ρ* < ∞.
σρ*(σ)
Then from (2.5), we have
(2.7) n n) ≤ ρ* (ρ*T*) 1/(ρ*-1)
.
log ӏanӏ-1
ρ*-1
In all our further discussion, we shall denote (ρ*/ ρ*-1)( ρ*-1)
by the constant K. Then from (2.7) we have
(2.8) ӏanӏ < exp [ – λn. (λn) ],
{K.ρ*(T*+ϵ)}1/(ρ*-1)
Where ϵ > 0 is arbitrary and n > n0. Now for each f ϵ X, let us define
ӏӏ f ӏӏ q = ӏ an ӏ) exp [ – λn. (λn) ],
{Kρ*(T*+1/q)}1/(ρ*-1)
Where q = 1, 2, 3, .............. . In view of (2.8), ӏӏ f ӏӏ q exists and for q1 ≤ q2,
ӏӏ f ӏӏ q1 ≤ ӏӏ f ӏӏ q2. This norm induces a metric topology on X.
We define
λ(f, g) = q
ӏӏ f-g ӏӏ q .
1 + ӏӏ f-g ӏӏ q
The space X with the above metric λ will be donated by Xλ. Now we prove
Theorem 1 –: The space Xλ is a Fre’chet space.
Proof –: It is sufficient to show that Xλ is complete. Hence, let { fα } be a λ-Cauchy sequence in X. Therefore,
for any given ϵ > 0 there exists n0 = n0(ϵ) such that –:
ӏӏ fα - fβ ӏӏ q < ϵ V α, β > n0, q ≥ 1.
Denoting fα(s) = n(α) es.λn
, fβ(s) = n(β) es.λn
, we have therefore
(2.9) ӏ an(α) – an(β) ӏ ). exp [ λn. (λn) ] < ϵ
{Kρ*(T*+1/q)}1/(ρ*-1)
For α, β > n0 , q ≥ 1. Hence we obviously have
ӏ an(α) – an(β) ӏ < ϵ v α, β > n0 , i.e. { an(α) } is a Cauchy sequence of
Complex Numbers for each fixed n = 1, 2, 3 ........... .
Now letting β → ∞ in (2.9), we have for α > n0 ,

(2.10) ӏ an(α) – an ӏ ). exp [ – λn. (λn) ] < ϵ.
{Kρ*(T*+1/q)}1/(ρ*-1)
Taking α = n0 , we get for a fixed q,
ӏ an ӏ.exp [ λn. (λn) ] ≤ ӏ an
(n
0)
ӏ.exp [ λn. (λn) ] + ϵ.
{Kρ*(T*+1/q)}1/(ρ*-1)
{Kρ*(T*+1/q)}1/(ρ*-1)
Now, f(n
0)
= n
(n
0)
. e s.λ
n ϵ Xα , hence the condition (2.8) is satisfied. For
arbitrary p > q, we have
ӏ an
(n
0)
ӏ < exp [ – λn. (λn) ] for arbitrarily large n.
{Kρ*(T*+1/p)}1/(ρ*-1)
Hence we have
ӏ an ӏexp[ – λn. (λn) ] < exp[ – λn. (λn) 1 – 1 ] + ϵ
{Kρ*(T*+1/q)}1/(ρ*-1)
{Kρ}1/(ρ*-1)
(T*+1/q)}1/(ρ*-1)
(T*+1/p)}1/(ρ*-1)
Since ϵ > 0 is arbitrary and the first term on the R.H.S. → 0 as n → ∞, we find that the sequence {an} satisfies
(2.8). Then f(s) = n esλ
n belongs to Xλ.
Now, from (2.10), we have for q = 1, 2, 3 ........... , ӏӏ fα - f ӏӏ q < ϵ. Hence,
λ (fα , f) = q
ӏӏ fα-f ӏӏ q ≤ ϵ q
= ϵ < ϵ.
1 + ӏӏ fα-f ӏӏ q (1 + ϵ) (1 + ϵ)
Since the above inequality holds for all α > n0 , we finally get fα→f where f ϵ Xλ. Hence Xλ is complete. This
proves theorem 1.
Theorem 2 –: A continuous linear functional ψ on Xλ is of the form
ψ (f) = ncn if and only if
(2.11) ӏ cn ӏ ≤ A. exp [ λn. (λn) ]
{Kρ*(T*+1/q)}1/(ρ*-1)
For all n ≥ 1, q ≥ 1, where A is a finite, positive number, f = f(s) = n . e s.λ
n and λ1 is sufficiently large.
Proof –: Let ψ ϵ X’λ. Then for any sequence { fm } ϵ Xλ such that fm → f, we have ψ( fm) → ψ( f ) as m → ∞.
Now let –:
f(s) = n . e s.λ
n ,
where a’ns satisfy (2.8). Then f ϵ Xλ. Also, let
fm(s) = n . e s.λ
n
Then fm ϵ Xλ for m = 1, 2, 3 .......... . Let q be any fixed positive integer and let 0 < ϵ < 1/q.
From (2.8), we can find an integer m such that
ӏ an ӏ < exp [ – λn. (λn) ] , n > m
{Kρ*(T*+ϵ)}1/(ρ*-1)

Then,
ӏӏ f – n . e s.λ
n ӏӏq = ӏӏ n . e s.λ
n ӏӏq
= ӏ an ӏ) exp [ λn. (λn) ]
{Kρ*(T*+1/q)}1/(ρ*-1)
< exp [ λn. (λn) ] 1 – 1
{Kρ*}1/(ρ*-1)
(T*+1/q)}1/(ρ*-1)
(T*+1/p)}1/(ρ*-1)
< ϵ for sufficiently large values of m.
Hence, λ( f, fm) = q
ӏӏ f-fm ӏӏ q ≤ ϵ < ϵ.
1 + ӏӏ f-fm ӏӏ q (1 + ϵ)
i.e. fm → f as m → ∞ in Xλ. Hence by assumption that ψ ϵ X’λ , we have
m) = ψ(f).
Let us donate by Cn = ψ(es.λ
n). Then
ψ(fm) = an ψ(es.λ
n) = an Cn
Also ӏ Cn ӏ = ӏ ψ(es.λ
n) ӏ. Since ψ is continuous on Xλ , it is continuous on ӏӏ X ӏӏq for each q = 1, 2, 3 ............... .
Hence there exists a positive constant A independent of q such that
ӏ ψ(es.λ
n) ӏ = ӏ Cn ӏ ≤ A ӏӏ α ӏӏq , q ≥ 1,
where α(s) = es.λ
n . Now using the definition of the norm for α(s), we get
ӏ Cn ӏ ≤ A.exp [ λn. (λn) ] , n ≥ 1, q ≥ 1.
{Kρ*(T*+1/q)}1/(ρ*-1)
Hence we get ψ(f) = an Cn , where C’ns satisfy (2.11).
Conversely, suppose that ψ(f) = anCn and Cn satisfies (2.11). Then for q ≥ 1,
ӏ ψ(f) ӏ ≤ A . ӏ an ӏ) exp [ λn. (λn) ] ,
{Kρ*(T*+1/q)}1/(ρ*-1)
i.e. ӏ ψ(f) ӏ ≤ A. ӏӏ f ӏӏ q , q ≥ 1 ,
i.e. ψ ϵ ӏӏ X’ ӏӏq , q ≥ 1. Now, since
λ(f, g) = q
ӏӏ f-fm ӏӏ q ,
1 + ӏӏ f-fm ӏӏ q
therefore X’λ = ӏӏ X’ ӏӏq. Hence ψ ϵ X’λ.
This completes the proof of Theorem 2. Lastly, we give the construction of total sets in Xλ. Following [2], we
give –:
Definition –: Let X be a locally convex topological vector space A set E X is said to be total if and
only if for any ψ ϵ X’ with ψ(E) = 0, we have ψ = 0.
Now, we prove
Theorem 3 –: Consider the space Xλ defined before and let f(s) = n (es.λ
n),

an 0 for n = 1, 2, 3 ............ , f ϵ Xλ. Suppose G is a subset of the complex plane having at
least one limit point in the complex plane. Define, for μ ϵ G, fμ (s) = n eμ.λ
n).(es.λ
n). Then E = { fμ ; μ ϵ
G } is total in Xλ .
Proof –: Since f ϵ Xλ , from (2.7) we have
λn. (λn) = (λn)
log ӏ an.eμ.λ
n ӏ -1
log ӏ an ӏ -1/λ
n – R(μ)
≤ ρ* (ρ*T*)1/(ρ*-1)
, since R(μ) < ∞.
ρ* - 1
Hence, if we donate by Mμ(σ) = sup ӏ fμ (σ + it) ӏ , then from (2.6),
-∞ < t <∞
log Mμ (σ) ≤ T* < ∞.
P*(σ)
Therefore, fμ ϵ Xλ for each μ ϵ G. Thus E Xλ . Now, let ψ be a linear continuous functional on Xλ and
suppose that ψ(fμ) = 0. From Theorem 2, there exists a sequence {Cn} of complex numbers such that
ψ(g) = bn Cn , g(s) = bn es.λ
n ϵ Xλ ,
where
(2.12) ӏ Cnӏ < A.exp [ λn. (λn) ] , n ≥ 1, q ≥ 1,
{Kρ*(T*+1/q)}1/(ρ*-1)
A being a constant and λ1 is sufficiently large.
Hence,
ψ( fμ ) an Cn eμλ
n = 0 , μ ϵ G.
References
[1]. K.N. Awasthi : A study in the mean values and the growth of entire functions, Ph.D. Thesis, Kanpur University, 1969.
[2]. T. Hussain and P.K. Kamthan: Spaces of Entire functions represented by Dirichlet Series. Collect. Math., 19(3), 1968,203-216.
[3]. P.K. Kamthan: Proximate order (R) of Entire functions represented by Dirichlet Series. Collect. Math.,14(3), (1962),275-278.
[4]. P.K. Kamthan: FK –spaces for entire Dirichlet functions. Collect.Math., 20(1969), 271-280.
[5]. Q.I. Rahman : On the maximum modulus and the coefficients of a Dirichlet Series. Quart. J. Math. (2), 7(1956),96-99.
[6]. J.F.Ritt: On certain points in the theory of Dirichlet Series. Amer. J. Maths., 50(1928), 73-86.
[7]. G.S. Srivastava: A note on proximate order of entire functions represented by Dirichlet Series. Bull De L’ Academie polonaise Des
Sci 19(3)(1971),199-202.

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On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet Series