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ISSN 2229 – 5046
International Journal of Mathematical Archive- 8(7), July – 2017 42
RELATIVE L-RITT ORDER OF ENTIRE DIRICHLET SERIES
DIBYENDU BANERJEE*, SIMUL SARKAR
Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal, India.
(Received On: 14-06-17; Revised & Accepted On: 06-07-17)
ABSTRACT
W
e introduce the idea of relative L-Ritt order of entire Dirichlet series with respect to a meromorphic function and prove sum and product theorems and a theorem on derivative.Keywords: Entire Dirichlet series, Relative order, L-Ritt order.
AMS Subject Classification: 30B50, 30D99.
1. INTRODUCTION AND DEFINITIONS
For entire function
f
letF
(
r
)
=
max{|
f
(
z
)
|:|
z
|
=
r
}.
Iff
is non constant thenF
(
r
)
is strictly increasing and a continuous function ofr
and its inverse)
,
0
(
)
|,
)
0
(
(|
:
1
∞
→
∞
−
f
F
exists andlim
−1(
)
=
∞
.
∞
→
F
R
R
In 1988, Bernal [1] introduced the definition of relative order of
f
with respect tog
denoted byρ
g(
f
)
, as)
(
)
(
:
0
inf{
)
(
µ
µρ
gf
=
>
F
r
<
G
r
for allr
>
r
0(
µ
)
>
0
}.
Let
f
(
s
)
be an entire function of the complex variables
=
σ
+
it
defined by everywhere absolutely convergent Dirichlet series∑
∞=1
n s n
n
e
a
λ (1.1)where
0
<
λ
n<
λ
n+1(
n
≥
1
)
,λ
n→
∞
asn
→
∞
anda
ns
'
are complex constants.
Let
F
(
σ
)
=
l
.
u
.
b
{|
f
(
σ
+
it
)
|,
−∞
<
t
<
∞
}.
Then the Ritt order [7] off
(
s
)
, denoted byρ
(
f
)
is given byσ
σ
ρ
σ
)
(
log
log
sup
lim
)
(
f
F
∞ →
=
=
inf{
µ
>
0
:
log
F
(
σ
)
<
exp(
σµ
)
for allσ
>
σ
(
µ
)}.
In [5] Lahiri and Banerjee introduced relative Ritt order as follows.
The relative Ritt order of
f
(
s
)
with respect to an entireg
(
s
)
is defined by)
(
)
(
log
:
0
inf{
)
(
µ
σ
σµ
ρ
gf
=
>
F
<
G
for all largeσ
}
where
G
(
r
)
=
max{|
g
(
s
)
|:|
s
|
=
r
}.
Let
L
=
L
(
σ
)
be a positive continuous function increasing slowly i.e.,L
(
a
σ
)
≈
L
(
σ
)
asσ
→
∞
for every constanta
.
Corresponding Author:
Dibyendu Banerjee*,
Then L-Ritt order [4] of
f
(
s
)
is defined as follows:.
)
(
)
(
log
log
sup
lim
)
(
σ
σ
σ
ρ
σ
L
F
f
L
∞ →
=
In 2014, A. Kumar and A. Rastogi [4] introduced relative L-Ritt order of an entire Dirichlet series as follows.
The relative L-Ritt order
ρ
gL(
f
)
off
(
s
)
with respect tog
(
s
)
is defined as.
)
(
)
(
log
sup
lim
)
(
1
σ
σ
σ
ρ
σ
L
F
G
f
L g
− ∞ →
=
At this stage it therefore seems reasonable to define suitably the relative L-Ritt order of entire Dirichlet series (1.1) with respect to a meromorphic function and to enquire its basic properties.
First we define characteristic function of an entire function
f
(
s
)
defined by everywhere absolutely convergent Dirichlet series (1.1) by.
|
)
(
|
log
2
1
)
(
σ
θ
π
σ
π θπ
d
e
f
T
f∫
i− +
=
Clearly
T
f(
σ
)
≤
log
+F
(
σ
).
The following definition is now introduced.
Definition 1.1: The relative L-Ritt order
ρ
gL(
f
)
off
(
s
)
with respect to a meromorphicg
(
s
)
is defined as[
σ
σ
]
µσ
µ
ρ
L(
f
)
inf{
0
:
T
f(
)
T
g(
)
L
(
)
g
=
>
<
for all largeσ
}
where
T
g(
σ
)
is the Nevanlinna Characteristic function ofg
(
s
).
Note 1.1: It is clear that
[
]
.
)
(
)
(
log
)
(
log
sup
lim
)
(
σ
σ
σ
ρ
σ
T
L
T
f
g f L
g
∞ →
=
Definition 1.2: A non constant meromorphic function
g
(
s
)
is said to have the property (B) if for anyn
>
1
and large,
σ
T
g(
n
σ
)
=
O
(
T
g(
σ
)).
2. SUM AND PRODUCT THEOREMS
In this section we assume that
f
1,
f
2 etc., are entire functions ofs
defined by everywhere absolutely convergentordinary Dirichlet series
∑
∑
∞ = ∞
=1 1
,
n s n
n s n
n
b
n
a
etc. The product of two such series is considered by Dirichlet product
method which is also everywhere absolutely convergent {see [3], pp 66}.
Theorem 2.1: Let
f
1 andf
2 be entire functions having respective relative L-Ritt ordersρ
gL(
f
1)
andρ
gL(
f
2)
with respect to meromorphicg
.
Then (i)
ρ
gL(
f
1±
f
2)
≤
max{
ρ
gL(
f
1),
ρ
gL(
f
2)}
and (ii)ρ
gL(
f
1f
2)
≤
max{
ρ
gL(
f
1),
ρ
gL(
f
2)}.
Proof: We may suppose that
ρ
gL(
f
1)
andρ
gL(
f
2)
are both finite, because if one ofρ
gL(
f
1)
,ρ
gL(
f
2)
or both are infinite, the inequality are evident.© 2017, IJMA. All Rights Reserved 44 For arbitrary
ε
>
0
and for all largeσ
, we have[
σ
σ
]
ρ ε[
σ
σ
]
ρ εσ
<
1+≤
2+1
(
)
T
(
)
L
(
)
T
(
)
L
(
)
T
f g g and(
)
[
(
)
(
)
]
.
2 2
ε ρ
σ
σ
σ
<
+L
T
T
f gNow for all large
σ
,
T
f1±f2(
σ
)
≤
T
f1(
σ
)
+
T
f2(
σ
)
+
log
2
<
[
σ
σ
]
ρ2+ε)
(
)
(
3
T
gL
<
[
T
(
σ
)
L
(
σ
)
]
ρ2+3ε.
g
So
ρ
gL(
f
1±
f
2)
≤
ρ
2+
3
ε
.
Since
ε
>
0
is arbitrary,)}
(
),
(
max{
)
(
f
1f
2 2 gLf
1 gLf
2L
g
ρ
ρ
ρ
ρ
±
≤
=
which proves (i).
For (ii), since
T
f1f2(
σ
)
≤
T
f1(
σ
)
+
T
f2(
σ
)
<
[
σ
σ
]
ρ2+ε)
(
)
(
2
T
gL
<
[
T
(
σ
)
L
(
σ
)
]
ρ2+2ε.
g
So
ρ
gL(
f
1f
2)
≤
ρ
2+
2
ε
.
Since
ε
>
0
is arbitrary,)}.
(
),
(
max{
)
(
f
1f
2 2 gLf
1 gLf
2L
g
ρ
ρ
ρ
ρ
≤
=
3. RELATIVE L-RITT ORDER ON DERIVATIVE
Theorem 3.1: Let
f
be an entire function defined by (1.1) andg
is transcendental such thatg
andg
′
satisfy property (B). Thenρ
gL′(
f
)
=
ρ
gL(
f
).
We need the following lemmas.
Lemma 3.2 [6]: Let
g
be a transcendental meromorphic function. Then)}
2
(
{
)
2
(
2
)
(
σ
gσ
gσ
g
T
o
T
T
′≤
+
for all large values ofσ
.
Lemma 3.3 [2]: Let
g
be a meromorphic function. Then for all largeσ
,
}
log
)
2
(
{
)
(
σ
<
g′σ
+
σ
g
C
T
T
where
C
is a constant which is only dependent ong
(
0
).
Proof of the theorem: From Lemmas (3.2) and (3.3) we have for all large
σ
,
)
2
(
)
(
σ
1 gσ
g
K
T
T
′<
(3.1) andT
g(
σ
)
<
K
2T
g′(
2
σ
)
(3.2)where
K
1andK
2 are positive constants.For arbitrary
ε
>
0
and for all largeσ
,
[
(
)
(
)
]
.
)
(
σ
≤
′σ
σ
ρ′(f)+ε gf
L g
So,
log
( )
(
'( )
)
L
f g
T
σ
<
ρ
f
+
ε
log
K
1+
log
T
g(2 ) log ( )
σ
+
L
σ
, using (3.1)
=
(
ρ
gL′( )
f
+
ε
)
log( (
O T
g( ))) log ( )
σ
+
L
σ
+
O
(1)
, sinceg
has the property (B)=
(
ρ
gL′( )
f
+
ε
)
[log(
T
g( ) ( ))
σ
L
σ +
O
(1)]
Therefore,
[
]
(
)
.
)
(
)
(
log
)
(
log
sup
lim
ρ
ε
σ
σ
σ
σ→∞
T
L
≤
′f
+
T
Lg g
f
Since
ε
>
0
is arbitrary,).
(
)
(
f
gLf
L
g
≤
ρ
′ρ
(3.3)Also for arbitrary
ε
>
0
and for all largeσ
[
(
)
(
)
]
.
)
(
σ
<
gσ
σ
ρ (f)+εf L g
L
T
T
So using (3.2) and since
g
′
has property (B), we have(
)
log
T
f( )
σ
<
ρ
gL( )
f
+
ε
log(
T
g′( ) ( ))
σ
L
σ
+
O
(1) .
Therefore,
[
]
(
)
.
)
(
)
(
log
)
(
log
sup
lim
ρ
ε
σ
σ
σ
σ→∞
T
′L
≤
f
+
T
Lg g
f
Since
ε
>
0
is arbitrary,).
(
)
(
f
Lf
g L
g
ρ
ρ
′≤
(3.4)Hence from (3.3) and (3.4)
).
(
)
(
f
gLf
L
g
ρ
ρ
′=
4. FINITENESS OF
ρ
gL(
f
)
Definition 4.1: Let
f
be entire andg
be a meromorphic function which is not transcendental.Let
α
=
inf
µ
such that[
σ
σ
]
σ
σ
σ
σ µd
L
T
L
T
g f∫
∞ + 0 1)
(
)
(
)
(
)
(
,
σ
0≥
σ
0'>
0
converges.Lemma 4.1: If
[
σ
σ
]
σ
σ
σ
σ µd
L
T
L
T
g f∫
∞ + 0 1)
(
)
(
)
(
)
(
is convergent then
[
]
0
)
(
)
(
)
(
lim
=
∞ → µσ
σ
σ
σ
L
T
T
g fwhere
0
<
µ
<
∞
.
Proof: Given
ε
>
0
,
there is a numberσ
'(
ε
)
≥
σ
0 such that[
]
∫
∞ +<
σ µε
dt
t
L
t
T
t
L
t
T
g f 1)
(
)
(
)
(
)
(
whenever
σ >
σ
'(
ε
)
and so
[
]
∫
σ +<
σ µ
ε
2 1)
(
)
(
)
(
)
(
dt
t
L
t
T
t
L
t
T
g ffor
σ >
σ
'(
ε
)
.Since
T
f(
σ
),
T
g(
σ
)
andL
(
σ
)
are non-decreasing, we have forσ >
σ
'(
ε
)
[
]
+∫
>
σ σ µσ
σ
σ
σ
ε
12© 2017, IJMA. All Rights Reserved 46
[
]
.
)
2
(
)
2
(
)
(
)
(
1 +=
µσ
σ
σ
σ
σ
L
T
L
T
g f So,[
]
[
[
]
]
µµ
µ
σ
σ
σ
σ
σ
ε
σ
σ
σ
σ
)
(
)
(
)
2
(
)
2
(
.
)
(
)
(
)
(
)
(
1L
T
L
T
L
T
L
T
g g g f +<
i.e.,[
]
.
)
(
)
(
)
2
(
)
2
(
)
(
)
2
(
2
log
)
2
(
2
log
)
(
)
(
)
(
ε
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
µ µ
<
L
T
L
T
L
L
T
L
T
T
g g g g fSince
g
is not transcendental, soT
g(
σ
)
=
O
(log
σ
)
and henceT
g(
2
σ
)
=
O
(
T
g(
σ
))
and alsoL
(
2
σ
)
≈
L
(
σ
)
as.
∞
→
σ
So
[
]
0
.
)
(
)
(
)
(
lim
=
∞ → µσ
σ
σ
σ
L
T
T
g fThis proves the lemma.
Theorem 4.2: If
ρ
gL(
f
)
be the relative L-Ritt order off
with respect tog
andα
is defined by Definition (4.1), thenρ
gL(
f
)
is finite ifα
is finite.Proof: Suppose
α
is given.Then for arbitrary
ε
>
0
, the integral[
]
∫
∞ + + 0 1)
(
)
(
)
(
)
(
σ α ε
σ
σ
σ
σ
σ
d
L
T
L
T
g f converges.So by Lemma (4.1)
[
]
0
.
)
(
)
(
)
(
lim
+=
∞
→ α ε
σ
σ
σ
σ
L
T
T
g fThus for all sufficiently large values of
σ
[
σ
σ
]
ε
σ
ε α+
<
)
(
)
(
)
(
L
T
T
g fi.e.,
T
f(
σ
)
<
ε
[
T
g(
σ
)
L
(
σ
)
]
α+εi.e.,
log
T
f(
σ
)
<
log
ε
+
(
α
+
ε
)
log
[
T
g(
σ
)
L
(
σ
)
]
.
So,
[
]
.
)
(
)
(
log
)
(
log
sup
lim
α
ε
σ
σ
σ
σ→∞
T
L
≤
+
T
g f
Since
ε
>
0
is arbitrary,ρ
gL(
f
)
≤
α
and this proves the theorem.REFERENCES
1. Bernal, L., Orden relativo de crecimiento de funciones enteras, Collect.Math., 39 (1988), pp. 209-229.
2. Dai Chongi and Jin Lu, Number of deficient values of a class of meromorphic functions, Kodai Math J. 10 (1974), pp. 74-82.
3. Hardy, G. H. and Riesz, M., The general theory of Dirichlet series, Stechert-Hafner Service agency, New York and London (1964).
5. Lahiri, B. K. and Banerjee, D., Relative Ritt order of an entire Dirichlet series, Int. J. Contemp. Math. 5(44) (2010), pp. 2157-2165.
6. Lahiri, I., Generalised order of the derivative of a meromorphic function II, Soochow J. Math., 16(1) (1990), pp. 11-15.
7. Rahman, Q. I., The Ritt order of the derivative of an entire function, Annales Polonici Mathematici, 17 (1965), pp. 137-140.
Source of support: Nil, Conflict of interest: None Declared.
