I ntnat. J. Math. Math. Sci.
Vol. 2 (1979) 45-60
45
ON THE OVERCONVERGENCE OF CERTAIN SERIES
M. BLAMBERT and R. PARVATHAM
Institut Fourier
Mathmatiques Pures
Boite postale 116
38402 ST MARTIN D’HERES
FRANCE
(Received April 27, 1978)
ABSTRACT.
In this work, we consider certain class of exponential
series
with polynomial coefficients and study the properties of convergence of such
series. Then we consider a subclass of this class and prove ce[tain theorems
n the overconvergence of such a series, which allow us to determine the
conditions under which the boundary of the region of convergence of this
series is a natural boundary for the function
f
defined by this series.
KEY WORDS AND PHRASES. LC-Dirichletian element, L-Dirichletian element,
Convergence, Overconvergence.
AMS (MOS) SUBJECT CASSIFICATION (1920) CODES. $0A6, $0A64.
1. INTRODUCTION.
Le us consider the following LC-dirichletian element
If]
" Pn(x)mn
1
where
Pn (s)
j=O
(i.I)
exp-knS
anjS
an..j
are complex constants with
an,mnO
s=g+ir,
46
M. BLAMBERT & R. PARVATHAM
I
((,I) E
(kn)
and
is a sequence of complex numbers such that
(Iknl)
a D-sequence. That is to say
(l nl)
is a sequence of positive real numbers
satisfying
Ixxl < I21
0 <
<
Let
L
lim sup
An
Max
n
mn
n
lim sup
gn
n
LJ
g
-
n
=I s6
mn)
s
(1.4)
/n6 IN-J0}
Let us denote by
depea4/ng on
that
(1.3)
(1.5)
be the set of points of
01
(s)
(ni) Pn
(1.2)
/n6]N-[0}
Ilanl/6 (0,
and
Let
I%1
lim
n-
gd
(
which are zeros of
the derived set of
where
(nj)
n
and
and
IN-[0]
is an infinite subsequence of
dug
let
P (s)
is a closed set.
Let us suppose
is non empty. We put
Iog
6(n,s)
-
l(s)exp(-s)
for sufficiently large n
In his paper, using a technique similar to that used by M. B lamber[
and
J.
Simeon
enable us
o
[2], we prove two lemmas for a LC-dichleian elemen which
discuss Zhe proies of absolute convergence and uniform conver-
gence for (I. I) in
exclusively. Then we prove
Jentzsch’s heorem for
a L-dichletian elemen tha is for element of he type (I. I) where
positive real numbers satisfying
(1.2)
[(k n)
is a
D-sequence)
kn
are
and a heorem
on he overconvergence for a L-dichletian element.
2. MAIN RESULTS.
DEFINITION.
It is said that a function is sub-lipschitzian on an open
set, if it is lipschitzian on each compact subset of that open set.
47
OVERCONVERGENCE OF SERIES
LEMMA 1.
Let
be any compact subset of
Then the following
assertions are true.
(1)
(2)
and If there exists a
then the function
v
[1,2,..., mn]
_w
n’ nn’
[6w(So)
such that
<
(-
Then it is easy to see that
(l’2""’mn) = nj
{J
V
(],)
is the open disc centred at
ds,
where
dist
e
3
]0,[
so 6
is sub-llpschitzlan on
6
w
PROOF. Let
s
is lipschitztan.
n’ n>n’
<(R)
If
6 (n, s)
9s
the function
3
c-
ds,
and
and of radius
s
is the sequence of zeros of
Pn(S)
(Cnj)
(with its order of mul-
plicity is taken into account). More precisely let us show that,
v
-
e]0,eX[
Ge
Let
s6
subset of
d
v
B
n’ nn’
ds, e It
Let
s
indexed by
S,’
[j6(1,2,...,mn)
s
is evident that
e’9
]0,e-e[
G
where
=Znj’ds,e
the closure of Ge
e]0,e[
The set of discs
Hence we have
is an open covering of
on
is a compact
a finite subcoverlng
3
o(s1,
Let
s
implies that
and
d
U
s k) J=l
s’
sj,’
d
s,
s’
3
J*(1 ,...,k)
J(1,...,k)
n’(=nj)
DG
s’G
hence
Then
s’
k
t
j=l
d
which
d
sj,
Now
sJ
nn’ sd
Pn(S)
0
Sj,’
and hence
P (s)# 0
nMax[/j(1
which gives
n
s
Maxtl/j6(1
is arbitrary on
n
k)] j(1,2,...,k) sd
k)}
we have
Pn (s’) 0
As
s
is arbitrary on
]i
and
48
H. BLAHBERT & R. PARVATHAH
v
3
E]0,}[ n’(=n)
= mnj
k)
[jE(1
y
nan’ sg
ds,e]
From which we have
log
V
V
nn’(=n
(s,s)xx
Under the above conditions elated to
I(n,s)-(n,s’)
mn
B
i(s)l-log (s’)
Is-s’l
i
o
mn
[s-s’[
g
,s’-%],,
s and s’ with
n
+
log 1+
s
log i+
S-S.’
j=l
s’
1+
j=l
Is-s,+’
ls-s’[
as
su,1(1
mn
1+
O
3
V
6]0,K
V
+
lo(l+lS-S’) s-s’l
]m
1
mn Is-s’
eknl
sup
log(1 )
x
I(n -(n,s’i
1
I-’l
16(n,s)-6(n,s’)i
V
/x>0
1.
Puttin
,n
n’ n>n’ (s,s’)E}x
which proves the first pact of the lemma.
Now let
cE]0,X[
las
lim sup p
,n
n-*m
1+/
with
E]O,)f[
as
SoEG-g
.(So)
(s
and
where
Inf[p/E]0,eX[]
1+--
Hence
which completes the proof of the lemma.
Under the condition
(2)
of Lemma
1
6
is continuous on
(13-g
<(R)
OVERCONVERGENCE OF SERIES
cz
which implies that
49
-ge
is an open subset of
but
/wz
can have
several connected components.
LEMMA 2.
8" <
= Kc-uc 8’>8
v
/
v
a 6 R
Let
PROOF.
then
3
n’ n>n’ s6K
such that
be a compact subset of
and let
/*c @
n(S)exp(-knS)l < exp(-nl(a-8’))
(otherwise the lemma is trivial)
#
We can easily see that
/.a.
P (s’) 0
n
v
v
3
e6]0,dist(s,)[ n’(=ns,c n>n’ s’6ds,c
v
s6([;-
is the closed disc centred at
d
s,
where
When
and of radius
s
P (s’)
v
v
s6-* e6]0 K[
n’(=,
0
n
nan’ s’6d
Let us consider the compact subset
Hence
e
of
-e
As
s6E
dist(E ,g) > 0
we have from lemma i,
6(n,s)-6(n,s’)
3
’6]0,dist( ,)[ n’(=n ,) nan’ (s,s’)6
where
mn
I+
Pc’, n
v
n,
n 6(n,s)-6(n,s’)
e’ k
v
s6
s’
,
(n,s)
p,nS-S’
In particular,
6ds,
Pe’,nS-S’
and hence
(n,s’)
Fuher
V
’,n s-s’
mn
3
8’>8
<
’
and
(n,sl-s-s’(+ ,I
(n,s’l
nmax[
Since
is a compact subset of
a
we have
3
sh
V
’>8 e’] 0,dist(e,e )[ n’ n>n’ s’6dS,e
?
s6E
n’=,
5(n,s’) >
v 6(n,s’) > a
nan’
0-Is-s’l(l + ,)
finally
(2.1)
H. BLAMBERT & R. PARVATHAH
50
]0,[
is arbitrary in
where
and hence
is an open covering for
k
[3 d
3
s 6 d
3
s ’(...)
D ]
Using (2.1) for the particular pair
s,,
s
on
Further
1= 1 sj,e
D(Sl...sk
we have
indexed by
ds,
The set of discs
(sj,,s)
we have
6(n,s) >
B
] 0 ,diet (g*,[
’>8"
Let
Max[nsj,8,,
n
n’(=i,,8,,,
j6(l...k)] and
e’ <
Choosing
BB
B
n" hen"
e]O,
6(n,s) > a-
is any arbitrary point of
8’>8e
B
n"
g
and
v
v
n"
v
6(n,s) >
s6
and hence
v
Nc
8’>8
THEOREM 1.
v
B
n’ nn’
When
converges absolutely on
deend
6(n,s) > --8’
hc_. 8’>8" n’ nn’
v
does not
nan" sE]
v
B
and
8’
-
8e
is arbitrary and strictly greater than
8’
we have
dist(e’e
---2
s
cE]0,h[--
Is-s,,l(l+r,)
,
dist(, e) <
we have
w
As
[s-5 < e
as
e’] 0,dist(ee, )[ n" nan"
8’>8
=-
6(n s) > -s(l+ )
3
where
nan’
v
sN
8e <
2,L+Se
8’
(s) exp-knS
L<
we have
<
expe kn(-8’))
the LC-dirichletian element
and uniformly on any compact subset of
,L+8
PROOF
Let us suppose that 2),L+8
We know that
compact subset of 2) ,L+8
is non empty Let
B
Ko c 2)
o
Let
be a
If]
51
OVERCONVERGENCE OF SERIES
8’ 6 ]Se, c-L[
From Lemma 2 we have,
IPn(s)exp(-knS)l
3
v
n’ nan’ s6}
-8’ > L
where
exp[-Iknl(e-’)
Hence
IPn(s)exp(-knS)]
n=n’
<
<
n=n’
and the series on the right hand side is convergent which proves that
converges absolutely and uniformly on
subset of
/
e,
Since }o is any arbitrary compact
o
converges uniformly on any compact subset of
If]
L+8 e
and absolutely on
/)e,L+ e
REMARK 1.
Then
By the following method, we obtain a bigger set of absolute
3
V
Let
If]
convergence for
s6/)eL s>0
6e(s)
/)eL
be supposed to be non-empty and
> L+ s
-Log
IPn(s)exp(-knS)
n=n,
IPn(s)exp-k n sl
n>n’s
/)e,L+8 e
>
<
3
n’s nan’s
(L+s) Iknl
6(n,s) > L+ s
L<
and
Hence
Z exp[-(L+e s) ll n 11
n=n
s
s
and as the series on the right hand side converges, the series (I.I) converges
eL
absolutely on
REMARK 2.
6e(s)
where
< 0
and
(nj)
In this
if]
we have no restriction on
resu|t,
-
diverges on
6 e (s) <
3
06R o+
(-ge-/)eo
If
Hence
s 6
-ge-Ao
3
3
(nj
is an infinite subsequence of
IN-[0]
8e
then
6(nj
s)
0
we have
<-0
Therefore
[Pnj(S)exp(-ln.S)l] > exp([knj[) >
and which shows that
[f]
diverges on
convergence of the series (1 .I) in
/)eo c
-ge-eo
(B-g e
When
L
and divergence in
We do not discuss the property of convergence of the series in
From here onwards we consider a L-dirichletian element,
e
52
M. BLAMBERT & R. PARVATHAM
,
If]
P (s) exp(-k s)
n
1
(2.2)
n
where (k) is a D-sequence (here
n
DEFINITION. It
k
is said that a
are positive real numbers).
n
D-sequence
(n)
is of the type
(A)
if the following conditions are satisfied
i)
exp(-k s)
the Ditichlet series
(this gives that
n6IN-[0]
j=n
be its sum at the point
8 (s)
n
v>
ii)
0
]>0
8n pO
is easy to see that
L
Po
Let
(8)
n
where
8
n
"P 9 s
o
e n (s)
[s6(/>q]
P
=I exp(-S(kn-k))
(kn)
is a
(kn)
If the D-sequence
that
converges on
the sequence of functions (8 ) where
n
n
is bounded on
9 s
EXAMPLE.- If
o
s);
the sequence of functions
is bounded on
iii)
exp(-s(kj-kn ))
[s6 Ig>O
P
converges on
n
j=l
D-sequence and
is of the type
(kn)
(h)
is of the type
(A)
3
p>0
]
Inf(k-’l-k,n-,,
=P
then it
then we can easily show
0
Now let us prove Jentzsch’s theorem for L-dirichletian element. This
theorem for Dirichlet series with complex exponents was proved by
T.M. Gallie [3]. First let us consider the associated Dirichlet
IrA
1
An
where
series of
If]
An expt-k n s)
is defined by
(1.4). Let
Op
be the abscisse of pseudo convergence of
g
fA
lim sup
n-
P
when
L
0
apfA
ifA]
Then we know that
log An
kn
is the same as
fA
c
the abscisse of convergence of
IfA].
OVERCONVERGENCE OF SERIES
Let
n’
and
n
53
n’
be two natural numbers such that
(n,n’)
denote the set, indexed by
>
E
Let
n
n,n’
which are zeros of the
of points of
LC-dirichletian polynomial
n’
S
n,
E
let
=n JP-’(s) exp(-sk)
n’(S)
denote the union of all sets
En, n,
(n,n’)
corresponding to all pairs
E
be the set formed by the points which are zeros for an infinity of
d
where E
(s)
Let us put E
is the derived
polynomials S
n,n
It is evident that E g and
E
set of E
is a closed subset of (
and
EduE
E
g
D
Ew
and hence
-g
(which implies
@)
Then we have
(k)
<
o
c
Let us suppose that the theorem is not true. Then there exists
S
v
n’>n>no
s6d(b,p)
n,n
IPn (s)exp(-kn s)
3
V
n(=ns,)
n>n’o
V
N
-g) and a disc
([;-g such that
included in
8’>*
(Fr(2>o)
(-g) c
centred at
< 8’
mn
lexp(-knS)
s6d(b,p)
have
n’(=n
0
C
we get
V
nan
Let
S
O’
V
sd(b,p)
I
n ,n
I
n(S)]
n>n
I/k n
(s)
=nl
>
8
and put
and hence
Hence putting n
o
fA
c
we
Max(no,n’o,n")o
0
IPn(s)exp(-knS)l
n
’
< A exp(k n ). From the definition of
n
A n < exp(o’k n)
v
3
fA
0" >0"
of radius
and
Let us take a certain
8’ Log[l+sup[ Isl/s6d(b,p)]] -Inf[o/s6d(b,p)]
IPn(s)exp(-k n s)
b
,(s) 0
An(l+Is I)
(mn/k n)
E
d(b,p)
(Fr(.o)N
b 6
Ve have
i’
(79
is of the type
n
PROOF.
3
no
n>n’o
@
then we have
> 0
m
-E
vVe suppose in what follows that
w<
point
[S n
g
Vhen the D-sequence
THEOREM 2.
and
D
P(s)exp(-k.s)
< exp((m+o’)k
and
is defined to be equal to
n)
v
s6d(b,p)
T
n i’
exp((I/kn) iOg
Sn
n(S)
l’n
(S
(s))
n
I’
n
(s))
where
I/kn
54
M. BLAMBERT & R. PARVATHAM
Im Log S n n(S) 6 ]-T,r] For each integer n >n the function
I
T
(s) is holomorphic on d(b,p)
We have
T
d(b,p)
9 s
n I ,n
n ,n
n
i/k n
i/k
v
I( Pj(s)exp(-ls)) n {exp(k n (w+o’) +Iogn)]
n(S)l
j=n I
I’
s6d(b,p)
ITn
log n
exp(w+o’) exp()
n
(k n)
Since
(A) which implies
is of the type
(Tnl
Hence the sequence of functions
mal on
d(b,p)
Let
extract’ed
From any
(T,,** i, n
subsequence of
which converges uniformly on
0
we have
limexp(-lIn)
n
n-==
is bounded and hence nor-
nnl
n
d(b,p)
be a compact subset of
L
Int D2)
#
wo
we can extract a subsequence
such that
E and the limit function is holomorphlc on the
Int
Let
E
1
lim T
n- n I ,n
v
S6K
d(b,p)
Now
1
N.,o
such that
InthNIntE1
.
is a compact subset of
[J
Then the subsequence extracted from the arbitrarily extracted sub-
sequence of
morphic in
(T,,
converges uniformly on
’n
Int(EiLJl)
one at each point of
K UE1
El
This results that the sequence
bitrary compact subset of
lim
v
e>0 n’l(=ns
and hence
T
n I’ n
d(b,p) N
(s)
(([;-ge-eo)
v
o
(Tnl,n)
conver-
1
is any arbitrary compact subset of
sd(b,p) ns 6 d(b Q) f]
O
to a limit function holo-
EUE1 and takes
Hence the limit function takes the value
ges to the same limit function on
As
UE
and continuous on the boundary of
the value one at each point of
Let
d(b,p)
be a compact subset of
Then we have
i.
,o
such that
d(b,p)
and
Int K [ Int E
1
1
Then
n
L pj(s) exp(-xjs)
,e)>nl n>n’l J=nl
E1
< (l+e)
kn
is any ar-
we have
OVERCONVERGENCE OF SERIES
IPn(So )exp(-%so)l
V
n>n
55
Snl ,n(So) -Snl ,n-I (So)
< z(i+4
kn
which gives
IPn(So)exp(-knSo)
Log
k
6.(s o
so 6
is arbitrary.
/).oI-*
Log(l+)
kn
n
as
m 0
Lo9 2
>
Hence we arrive at a contradiction that
which establishes the result.
Finally, let us prove a theorem on the overconvergence of
If]
defined
by (7). Before proving the theorem let us note that
Let
REMARK ’3.
D-sequence of the type
If
s
6
mh
sl/sE]
n
ciently large
Lg
(A)
As
Log An
k
-;
A c
>0
8"
<
-gc
mA
.-
c
o
and a
lim
e
m
+ __n
k
n
with
we have
I+Loi.I+A
#
8"
sequence
IPn(s)exp(-k n s) An(l+Is l)mnexp(-ok n ).
An(l+mA )mnexp(mAkn) where
compact set
mh is finite for suffiLog(l+mA)+m
n
-8*Log(l+mA)-m A
THEOREM 3. -When
and
be a
we have
kn
Hence
is a
A
IPn(S) exp(-kn s)
6.(s)
and (k n)
-6
We have
IPn(s)exp(-knS)
then
sup[
be any compact subset of
A
A c
(k n)
o
-c
c
8*Log(l+m A)
m
A
If
-8"
is a
D-sequence of the type
if there exist an infinite subsequence
of strictly positive numbers
(8)
(n)
(h)
;6
8"
<
IN, of IN-J0]
such that
+=
and
V
kn
+i
> (1+{9)k n
(2.3)
56
M. BLAMBERT & R. PARVATHAM
[Snv(S)}
then the sequence
ges at each point
with
f
Snv(S)
j=l
P.(s)exp(-kjs),j
conver-
s
If]
defined by
PROOF.
manner-
n
of any open simply connected subset (whose intersection
in non empty) of an open set included in (I;-g* in which the
)*,8*
function
vEIN, where
is holomo[phic.
Let us choose 3 bounded domains
A1
A2
A3
A2
holomorphic. Furthe
let
I f
the function*
([;-g*
A3
-g*
in an open subset of
and A
,i 2
in which
Fr(A I)
Fr(A 2)
Fr(A
and
in the following
3
and
A3
is included
If]
defined by
3)
is
satisfy a condition of
Hadamard’s type, namely
Log.M
3
bE] 0,1[
where
2
g
E( ,2,3)
It is easy to see that
> 8
A
aA
1
+ (1-b) Log M 3
3
Let us cons ide the set
c
A
is non empty and is an interval.
A
a
1
Max[ If(s) I/sEFr(Ai)]
M
V
b Log M
v
e>0
and
/ aA -e
1
Inf[5(s)IS6l)
D
A1
Let
CZA1 sup
Then
We can easily show that,
which implies that
ai
is a finite number.
Hence
1
from lemma 2,
8’E]B*’aAI
hence for
j=n+l
n
>
n
nl
n>n
IPn(s)exp(-k n s)
sEFr(AI
1
IPj(s) exp(- kjs)
<
j=n+l
exp[-k (a A
exp[-Xn+l(CA 1-’)]
Since
(k n)
is a
D-sequence of the type
a finite number strictly positive
lexp[-(k
V
nEIN
exp(-kn(a A -8))
<
j=n+l
thus we have for each
-k
B(8’)
n+l )s[
nn 1
(A)
8*)]
j=n+l
and
such that
B(’)
where
exp[-( A
-’)(kj-k n+l )}
a A -8’ > 0
1
es
ZA-8’
there exists
57
OVERCONVERGENCE OF SERIES
IPj(s) exp(-kj s)
j=n+l
[66R
12
Now let
v
Let
Isl/s63
sup[
mA 3
nan’
>
mn
kn
-cfA
(s)
Log A n
>
in
k
(1+IS I) +0
Log
n
(l+mA3) mA3
Log(l+mA3) mA3
with
c
3
and is an interval in R
which shows that
12
m
._._n Log
Then
LogAn
6(n,s)
We have
-
5(n,s)
n’(=n s) nan’
(2.4)
B(8’)exp{-kn+l(CAl- s’)}
I/0D a
3
S63
<
i
/a
*Log(l+m53) mA3
Then
c < -o c -8
supI 2
Let
a53
and hence
e>0 /’ah3-e 53
3
which implies that
Inf[5,(s)Is
We can easily show that
a finite number. Once again, from lemma 2, we get
cA3
B’>*
3
n
2
n>n2
sFr(A
3)
a53
is
< exp[
IPn(S) exp(-knS)
which gives
n2-1
n(n 2)
V
s6Fr(A3)
j=l
IPj(s)exp(-kjs)l
j=l
n2-1
n
IP(s)exp(-kjs)l+ J=n 2
n
=n 2
Let us choose
cases.
Case 1.
n
where
’
>
If
such that
CA3-8’
> 0
A3- ’
0
Now we examine the two
then
n
exp(-kj(A3-’))= eXPn(aAf’))
B"(8’)
#
exp[-(a A
is the sum of the series
j=O
-’)(k?kn)} < B"(’)exp(kn(CA3-’))
exp(-2(A--8’)k j)
3
H. BLAMBERT & R. PARVATHAM
58
Case 2.
A3-B’
If
<0
then
n
n
n
l=n2
i=n 2
l-n 2
(kn)
Since the D-sequence
B’(8’)
strictly positive
n
j=l
(^) there exists a finite number
such that
exp-(kn-k j) laA3-8’}]
V
n61N-[0]
is of the type
B’(8’)
which implies that
n
j=n
2
exp(-kj(A3-8’)) B’(8’)exp(knl%3-8’l)
Max[B’(8’),B"(’)]
B’"(8’)
On putting
we have
n
=n
B’" (8’)exp(k
exp(-kj(t3-8’))
2
n
ICA;8’ I)
(2.5)
Using the generalized form of Hadamard three circle theorem
3
b610, I[
bLogM
LogM2,
1,
+ (1-b)LogM
[4] we have
3,
where
Max[
M.
IRn (s) I/s6Fr(hi)
1,2,3
with
R n (s)
f(s)-
Xa
j=l
From (2.4) we have for
Pj(s)exp(-kjs)
n
n
>
%;
M
I,
because of
B
)exp[-(kn+l)(l-8 )]
B(8
(2.3)
Max[
Let
B’ (8’)
o
( -’)]
)exp[-(l+)kn--l
On putting
lf(s)l/s6Fr(h3)
we have from (2.5) for
M3,
< B(8
n a n
B o+B’’(’)exp(k
Max(Bo
B"(8’)
+
M)
j=l
5(s)exp(-kJs)l/s6Fr(h3)
2
n’ )
Then for
n a n
2
(2.6)
OVERCONVERGENCE OF SERIES
M
3,
,
B’o (’) exp(knlaA 3- 8’l)
(2.8)
Then using (2.7) and (2.8) in (2.6)we get, for
Log
Since
59
b Log B(’)+Cl-b)Log B’
o
n
max[nl,n2}
()+I-b(l+%)(cA 1-’)+l-b)
)-lim
and hence
lim Log M 2
which proves the theorem.
When the polynomial
Pn (s)
reduces to a complex number
we
n,o
get the famous Ostrowaski’s theorem [I] for Dirichlet series. Our theorem
contains
en
Fr
/o
PROOF.
a point
p > 0
(S n)
(8 n)
of strictly positive numbers such that
kn+ 1
>
v
3
n’ nan’
and
f
is a singular point for
then
anS
mn
In theorem 3 if we replace (2.3) by the condition that
there exists a sequence
lim
n--
Pn (s)
G.L. Lunt’z theorem [5] as a particular case when
COROLLARY.
a
(l+en)kn
defined by (2.2). In particular if
is a natural boundary for
(Fr/o)-g
then each point of
(Fr
2o)c_
(-g
f
Let us suppose that the corollary is false. Then there exists
b 6 (Fr 9
wo
on which
f
converges on
)[
(E-g
and a disc
is holomorphic.
d(b,p)
d(b,p)
b
centred at
and of radius
As a result of theorem 3 the sequence
From remark 2
There exists necessarily points common to
[f]
diverges on
(E-e-/eo
and
;-g-o
d(b,p)
For these
points there is a contradiction which establishes the corollary.
ACKNOWLEDGENT.
The second author would like to thank French
Government for financial support.
REFERENCES
Legons sur les progr&s rcents de la thorie des sries
de Dirichlet, Gauthier-Villars, Paris, 1933.
1.
BERNSTEIN V.
2.
BLAMBERT M. and SIMEON J.
Sur une technique d’tude des proprits
de converqence des sries de Dirichlet
exposants complexes,
An. Fac. Ci. Univ. Porto, 56 (1973), 1-33.
60
M. BLAMBERT & R. PARVATHAM
GALLIE T.M.
Mandelbrojt’s inequality and Dirichlet series with
complex exponents, Trans. Amer. Mat. Soc. 90 (1959), 57-72.
GOLUSIN G.M. -Geometric theory of functions of a complex variable,
Providence, American Mathematical Soc. 1969, Translations of
Mathematical monographs 26.
LUNTZ G.L.
On the over convergence of certain series, Iz__v. An__p_n.
Armjan, S.S.R. See Fiz. Mat. Nauk. 15 (5), (1962), 11-26.
Laboratoire de Mathmatiques Pures
Institut Fourier
dpendant de l’Universlt Sclentifique et Mdicale de Grenoble
associ
C.N.R.S.
B.P. 116
38402 ST MARTIN D’HERES (France)