I nternat. J. Math. Math. Sci.
Vol. 6 No.
(1985) 181-187
181
ON THE RITT ORDER OF A CERTAIN CLASS OF FUNCTIONS
R. PARVATHAM and J. THANGAMANI
The Ramanuj an Institute
University of Madras
Madras-600 005, INDIA
(Received August 14, 1981)
ABSTRACT.
The authors introduce the notions of Ritt order and lower order to
functions defined by the series
I
(-n s)
f (s) exp
n
where () is a D-sequence and
n
f (s) are entire functions of bounded index.
n
KEY WORDS AND PHRASES.
order, entire functions of bounded index, Dirichlet
series.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1
30A16, 30A64.
INTRODUCTION.
Let us consider an M-dirichletian element:
{}:
Zn= 1
where (A
n)
f (s) exp
n
(-nS),
s
0
+ i%,
(o %) e
(i.i)
is a D-sequence (a strictly increasing unbounded sequence of positive
numbers) and f (s) are entire functions of bounded index (defined below).
Conver-
n
gence properties of such elements were discussed by J.S.J. Mac Donnell in his
doctoral dissertation [1] under the conditions
where m
n
is the index of f
n
lim
n+oo
log)tn n
m
0 and
lim
n/oo
-n
0
n
In this paper, we first study the convergence prop-
erties of these elements with less restrictions, namely,
L
lim
sup
n+oo
lim
-
log n
<
and
(1.2)
n
sup
m
n
<
(1.3)
n
As the functions defined by these series are unbounded in the half-plane, it is
not possible to define Ritt order directly.
However, by making use of functions
182
R. PARVATHAM AND J. THANGAMANI
defined by associated intermediate series, we introduce the notions of Ritt order
and lower order to these functions.
2.
MAIN RESULTS.
[2j.
DEFINITION 2.1.
An entire function f is said to be of bounded
index if there exists a non-negative integer N such that
If(k) (s)
max
If(j) (s)
J!
k!
0_<k<N
(f
(0)
(s)
f(s))
The least such integer N is called the index of f.
for all j and for all s.
We require the following lemma which shows that an entire function of bounded
index is of exponential type.
LEMLA 2.2. [3], [2].
Let f be an entire function of bounded index N.
f( k )(0)
{max
(N + i)
O<k<N
n
max
n
{X}:
A
1
exp (N
Is
+ i)
(2.1)
Y. a s j be an entire function of bounded index m
nj
n
j=O
Let f (s)
A
k
Then
{lanj I/j
0 1
m
n
If
max
(J) (0)
j
j!
mn
0 1
(2.2)
exp(-% s) the associated dirichletian element whose abscissa of
n
n
REMARK 2.3.
log A
k;
c
convergence is denoted by
k
n
lim
%
n
n
It can be easily seen from Lemma 2.2 that
fn(S)
<
Ifn (J)
max
exp (m
J
0<j -<m
n
Isl
+ i)
n
A
exp (m
n
THEOREM 2.4.
n
If 0 <-
<
+ i)
Isl
i, the region of absolute convergence of (I.I) is the
exterior of the hyperbola centre (k(l-
B2)-I
O) and eccentricity
B -I
contained
in the half-plane o > k.
PROOF.
Using Remark 2.3, we have
If n (s)
exp(- s)
n
<
A
n
exp(m + i)
n
Is
exp(- )
n
(2.3)
ON THE RITT ORDER OF CERTAIN CLASS OF FUNCTIONS
B
From the definitions of k and
n’
n >
n"
n
> 0
it follows that for
A
< exp (k
+ )% n
m
n
+
+ )n
n
n’
>- n"
i < (
183
and
Hence
max(n’, n")
n(e)
Ifn(S) exp(-%nS)
-<
exp(-%n(O
k
E)]s])
(6 +
E
and
-<
f (s) exp(-% s)
n
n()
n
exp (-% (O
k
n
n()
(6 + E)
g
Isl)
The series in the right hand side converges provided
Isl
o- kwhich is valid only if o > k and 0 _<
> 0
(2.4)
< i.
Thus any point in the region of convergence of (i.i) must satiny
(_
2((72
k)2
+ T2
> 0
which reduces to
k
(o-
2
2
i
62
2
I
2
k
>
2
(i
2
(2.5)
62)I
from which the theorem follows.
REMARK 2.5.
0 the M-dirichletian element converges in the half-plane
If
o > k (which coincides with the half-plane of convergence of the associated series
{X})
thus giving the result of MacDonnell
i] as a particular case.
Next we proceed to introduce the notions of Ritt order and lower order for
functions defined by (i.I).
We need the following lemmas.
Let the M-dirichletian element given by (i.i) converge absolutely on E a
D
{s
o
E C
(7
LEMMA 2.6.
O,
n
where
N
o
JR} denote the imaginary
Under the conditions o
is hoomorphic on
PROOF.
.
g
axis.
<
and 0 _<
c
,
we have E
a
and
,
Using (2.3) we have
{0} s
Do
i if o > 0 and
Ifn (s) exp(-EnS)
@(7
=-i if o < 0.
< A
n
exp[-
n
(i
m
n
+
In
i
(7 s_
[(7[
)]
and
184
R. PARVATHAM AND J. THANGAMANI
8 <
Since 0 <
s
-
s
[i- (8 + g)
given E > 0
’
n’(=n
’
n’
n >
s g
An
<
The function
{s
E
/o(I-
g
(B + g)
uniformly on each E
]a
>
to
>
as
o
where
.
n=n
qbn’}:
Let G be any open subset of
@G:
.
G
9
G
s
Let
and of width 2.
LEMMA 2.7.
(I
’)
IR x
PROOF.
o
Let
by its
Let
{n
Then
converges
fn(S) exp(-InS).
Yl[
00) Is
G}.
,
converges absolutely in G if o C
and
(O,B)
/]’[-
{s
B(I,Z
where
.
(2.6)
.
is continuous on
@G
Since G is arbitrary on
}
{o} converges
Now we put
sup{lqb(s’)I/o’ ->
_<
oo; further
can be continued analytically on the totality
denote its analytic continuation.
M
we extend
indexed by p on
(s) is hol]mrphic on G.
qb
O
O O)].
We put
(@n’
absolutely on each point of
of
80]
Ioi
fi-k-nd
is
( + E)
iTO of D
O
inf{o(l- (8 + )
G
The number
p}
0X
c
fn (s)exp (-INS)
O
exp[-O In(l
For any point (on the imaginary axis) s
limiting value as s
D
o
s e
B(I,)
is the horizontal strip with
Y1
as axis
Then
Under the conditions o X
and 0 -<
C
M
(O,B)
(TI’ )
is bounded on each point
be fixed
arbitraril
on
+
’
< 1
< 1 we have
o e IR and lira M
x ]+o
and
B
Then given
with 0 <
(o,B)
0
E’ > O,
8 < i.
We have
ON THE RITT ORDER OF CERTAIN CLASS OF FUNCTIONS
( + e)(l + ’))
0 o) > o(i
e o
185
(l
(2.7)
( + )(1 + ’));
as a result of(2o6)and (2.7)
(O,B) <
An,
(-og,(l
Xn, (o,(1- (
and hence
lim M
n
lim M
finally,
o(I
we put M(o
e,)
M
g
IR
,
{In,(s) I/o
{s
g
B(I,)/IO
(O,B)
_<
Max
compact set,
o
Max
-< o E,
6o)
+ ( + g) (i + [’));
> 0(i
s g B(I
^
},
<
{,(o(i
-
0 as O ->
(O,B)
( + )
+ e) (l+ ’))
0 as o ->
(o,B)
Further, we have
> -O
(B + g)(l + ’))
exp
M(og,)
I,E)
is holomorphic on the
Then we get
is finite.
+ (B + E)(i + g’)),
n’
As
M(oe,), ,(oe,(i-(B+)(I+.’))};
is a strictly decreasing function in IRand hence
+ g)(l + g’)] >
,([i
+
,(O(i
+ (B + g)(l + ’)) <
(equality holds for o
,
,
((i
(B + g)(l + g’))
if
(o(i
( + )(i + g’))
if O >
< 0
and
O,
O) with
As all M-dirichletian polynomials satisfy the two properties of the lemma, in
each horizontal strip B(%,)
o ->
M$(O,B)
hen
lim
TgEOREN
log
sup
is called the Ritt order of
XB s
b
called the lower order of
2.9.
M@(o,B)
lim
O.
We put
pB
0B
is bounded for each o e ]Rand hence the function
is decreasing on IRwith
DEFINITION 2.8.
Then
M@(o,B)
gnder
on B.
V
IRx
+
0
Let
(,B)
)tBqb
lim inf
log
X
pB
_<
< l, we have
and 0
c
OXR
and
% -< %X
R
+
log_o M*(O B)
on B.
the conditions
(i,)
+
log_ M
}.
186
R. PARVATHAM AND J. THANGAMANI
where
and
%
are respectively the Ritt order and lower order of
X in the whole
plane.
PROOF.
"V"
g
Proceeding as in Lemma 2.6 for 0 -<
0 n’(=ng)
Now denoting by
{n,}
and
"
+ ]
n’
{kn ,}
s e
and
,
I@n,(s)
B( I,)
< i, we have
< )i
n
Ioi
(O,B) -<
,
which gives
g’ E IR
and hence
-
Isl
defined by the elements
sufficiently large:
(B + e)
[O(I
by (2.6)
( + E)
[(7(1
the holomorphic functions on
we have for o negative with
M
B
0
Bn RXn
_<
eO)
As adding finite number of terms to a holomorphic function defined by a classical
Dirichlet series does not affect its Ritt order [4], we add
{,}
and then
PRXn
n=0
ORX.
Now
n’ (o,B)
M(,B)
M
V
_<
g
where
{:,}:
n’-I
A
n
exp
(-%n s)
to
o
+ M @n’
(O,B)
IR
n-i
n= 1
f (s) exp(-% s); then
n
n
o
V
(,)
x
+
PB
< max
(PBn ’, PB@n
0
Bn
since
n’
0.
Finally we have
(r,)
IRx
0
and similarly we can show:
V
(,)
x
+
%B -< %R
0
Now we are in a position to define the Ritt order and lower order of
plane
{.
in the whole
ON THE RITT ORDER OF CERTAIN CLASS OF FUNCTIONS
DEFINITION 2.10.
We put
P
sup {p
XR
sup
/(1,9.)
e IRx
B0
{IB /(yl,)
e IRx
]R
and
Then
OR
187
is called the Ritt order
of
on
and
on
IR
+}
o
is called the lower order
of
REFERENCES
i.
MAC DONNELL, J.S.J. Doctoral dissertation, The Catholic University of
America, (1957).
2.
SHAH, S.M. Entire unctions of bounded index, Proc. Amer. Math. Soc. 19,
(1968) i017-1022.
3.
HAYN, W.K. Differential inequalities and local valency, Pacific J. Math.
44, (1973) 117-137.
4.
BLAMBERT, M.
entieres
Sur la comparison des ordes boreliens et (R) des functions
d’une certaene classe, Bull. Sc. Math. 89, (1965) 103-125.