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On the singularities of analytic functions defined by L-Dirichletian elements
JOURNAL
OF MATHEMATlCAL
ANALYSIS
AND
APPLICATIONS
82,
292-294
On the Singularities of Analytic Functions by L-Dirichletian Elements M. BLAMBERT*
(1981)
Defined
AND R. PARVATHAM
Institut Fourier, B.P. 116-38402 Saint-Martin
&H&es, France
Submitted by R. P. Boas
In this note, using the Dzrbasjan-Mandelbrojt inequality, we locate the singularities of analytic functions defined by L-Dirichletian elements. Though Dzrbasjan remarks that his inequality can be used to locate the singularities of functions defined by the Dirichlet-Taylor series, it is the convergence theory we established earlier IM. Blambert and M. Berland. C. R. Acad. Sci. Paris 280 (1975) 263-266; M. Blambert and R. Parvatham, Ann. inst. Fourier 29 (1979), 239-2621 that leads us exactly to determine the natural boundaries of such functions, whereas Dzrbasjan is never concerned with the convergence of such a series, as he himself states.
Let us consider an L-Dirichletian element,
{f): c P,(s)exp(--A, s), s=a+it,(a,z)EIR2,
(1)
n-1
where P,(S) is a complex polynomial of degree m,, (A,,)? is a D-sequence (that is, a positive, strictly increasing, unbounded sequence) with upper density D* < 00. Let P,(S) = CJ’!!, a,,jsj, where an,m,,# 0; let Zn be the set of points of C which are zeros for P, and let 8m be the set of points of C which are zeros for an infinity of polynomials P,. Let us put 8 = 0, & and P* = Bd U cZ&, where Zd is the derived set of 8, and let us supposethat C - 8* # 0. Let us define
V
YE:-4’
6,(s) = lim inf n-cc
-LogP,(s)ew-4s)l I 1 AI
and CL*,?= {s E C - 8* 16,(s) > a}. We know that [ 1, 21 under the conditions D* < co (which implies trivially that lim,,, logn/A, = 0) and /3* = limsup,,,(m,/A,) < co, the series (1) * Deceased.
292 0022-247X/8 Copyright All rights
3/070292-03902.00/O D 1981 by Academc Press. Inc. of reproduction in any form reserved.
SINGULARITIES
OF ANALYTIC
293
FUNCTIONS
converges absolutely m 9*,, and diverges in C - 8* -G.+,, the closure of 9*0. Let us put
where G*O is
L(s) = no, (1 -;)m”+‘, n IYn il +-iE {O, l,..., m,} ! i 2. I
M,=Max
(where yU,i is the value at s = A,, of the ith derivative of the function (s - An)“‘n+i/L(s), which is defined by continuity at the point A,), y = lim sup n-t’x
i
LogM, 48
s t-+
1
and y+ = Max(0, y). Denoting by o* the mean upper density of the sequence (A,,) [4,5 1, we can prove the following: THEOREM
1.
Under the conditions
D* < co,
p* < co,
IYI < CfJ
and
Q*,o f a
we cannot obtain a holomorphic extension of the function g*O E) s + f(s) (where f(s) is the sum of the series (1) at the point s) to the open set
%-;,+U (UsEFrY,,-y+d(s,nR)) with R > o*. ProoJ Following the notations of Dzrbasjan [4], using the Dzrbasjan-Mandelbrojt inequality [5,4] and proceeding as in another of our papers ([ 3, see inequality (11.3)]), we get
where v is the excess function for Dzrbasjan sequence [4]! Mf(s,, nR) is the maximum modulus of the holomorphic extension off to the disc d(s,, xR) along a channel of width 27cR, with the Jordan arc joining the point s, to a point s, as its axis, such that d(s,, 7cR) c g*O. Hence
Now for s, E C - 8*, we have
294
BLAMBERT
AND PARVATHAM
We know that 13, see inequality (11.2)] lim sup
I
Log~ w, n
!
Hence, finally we have
Now let us supposethat there exists a point sbE Fr(G& ,_ y+-t) with E > 0, such that f admits a holomorphic extension to d(sb, xR) along a channel of width 271R with the Jordan arc joining sb to s, as axis, such that d(s,, nR) = %,o. By definition we have S,($,) = -(y’ + E) < -y+. But as a consequence of the result established above, we must have S,(sb) > --y+. Hence a contradiction establishesthe truth of the theorem. Now we also have COROLLARY 2. If yt = 0 and o* = 0, then Fr(.Y) is a natural boundary for the analytic function dejined by (1) in F’, where F is a connected component of C2*0. If Q*O has more than one connected component, then to each connected component there corresponds an analytic function dejked by (1) in that component.
ACKNOWLEDGMENT The second author wishes to thank the French government for financial support during the preparation of this work. REFERENCES 1. M. BLAMBERT AND M. BERLAND, Sur la convergence des ClCments L-dirichlttiens, C. R. Acad. Sci. Paris 280 (197.5), 263-266. 2. M. BLAMBERT AND R. PARVATHAM, Ultraconvergence et singularitts pour une classe de sbries d’exponentielles, Ann. Inst. Fourier 29 (1979), 239-262. 3. M. BLAMBER~ AND R. PARVATHAM, Sur les ordres infkrieur et supCrieur des fonctions entitres dttinies par des iliments L-dirichlittiens, to appear. 4. M. M. DZRBASJAN, Uniqueness theorems for analytic functions asymptotically representable by Dirichlet-Taylor series, Math. U.S.S.R. Sbornik 20 (1973), 603-649. 5. S. MANDELBROJT. Siries adhkrentes, rkgularisation des suites applications, Paris, 1952.
OF MATHEMATlCAL
ANALYSIS
AND
APPLICATIONS
82,
292-294
On the Singularities of Analytic Functions by L-Dirichletian Elements M. BLAMBERT*
(1981)
Defined
AND R. PARVATHAM
Institut Fourier, B.P. 116-38402 Saint-Martin
&H&es, France
Submitted by R. P. Boas
In this note, using the Dzrbasjan-Mandelbrojt inequality, we locate the singularities of analytic functions defined by L-Dirichletian elements. Though Dzrbasjan remarks that his inequality can be used to locate the singularities of functions defined by the Dirichlet-Taylor series, it is the convergence theory we established earlier IM. Blambert and M. Berland. C. R. Acad. Sci. Paris 280 (1975) 263-266; M. Blambert and R. Parvatham, Ann. inst. Fourier 29 (1979), 239-2621 that leads us exactly to determine the natural boundaries of such functions, whereas Dzrbasjan is never concerned with the convergence of such a series, as he himself states.
Let us consider an L-Dirichletian element,
{f): c P,(s)exp(--A, s), s=a+it,(a,z)EIR2,
(1)
n-1
where P,(S) is a complex polynomial of degree m,, (A,,)? is a D-sequence (that is, a positive, strictly increasing, unbounded sequence) with upper density D* < 00. Let P,(S) = CJ’!!, a,,jsj, where an,m,,# 0; let Zn be the set of points of C which are zeros for P, and let 8m be the set of points of C which are zeros for an infinity of polynomials P,. Let us put 8 = 0, & and P* = Bd U cZ&, where Zd is the derived set of 8, and let us supposethat C - 8* # 0. Let us define
V
YE:-4’
6,(s) = lim inf n-cc
-LogP,(s)ew-4s)l I 1 AI
and CL*,?= {s E C - 8* 16,(s) > a}. We know that [ 1, 21 under the conditions D* < co (which implies trivially that lim,,, logn/A, = 0) and /3* = limsup,,,(m,/A,) < co, the series (1) * Deceased.
292 0022-247X/8 Copyright All rights
3/070292-03902.00/O D 1981 by Academc Press. Inc. of reproduction in any form reserved.
SINGULARITIES
OF ANALYTIC
293
FUNCTIONS
converges absolutely m 9*,, and diverges in C - 8* -G.+,, the closure of 9*0. Let us put
where G*O is
L(s) = no, (1 -;)m”+‘, n IYn il +-iE {O, l,..., m,} ! i 2. I
M,=Max
(where yU,i is the value at s = A,, of the ith derivative of the function (s - An)“‘n+i/L(s), which is defined by continuity at the point A,), y = lim sup n-t’x
i
LogM, 48
s t-+
1
and y+ = Max(0, y). Denoting by o* the mean upper density of the sequence (A,,) [4,5 1, we can prove the following: THEOREM
1.
Under the conditions
D* < co,
p* < co,
IYI < CfJ
and
Q*,o f a
we cannot obtain a holomorphic extension of the function g*O E) s + f(s) (where f(s) is the sum of the series (1) at the point s) to the open set
%-;,+U (UsEFrY,,-y+d(s,nR)) with R > o*. ProoJ Following the notations of Dzrbasjan [4], using the Dzrbasjan-Mandelbrojt inequality [5,4] and proceeding as in another of our papers ([ 3, see inequality (11.3)]), we get
where v is the excess function for Dzrbasjan sequence [4]! Mf(s,, nR) is the maximum modulus of the holomorphic extension off to the disc d(s,, xR) along a channel of width 27cR, with the Jordan arc joining the point s, to a point s, as its axis, such that d(s,, 7cR) c g*O. Hence
Now for s, E C - 8*, we have
294
BLAMBERT
AND PARVATHAM
We know that 13, see inequality (11.2)] lim sup
I
Log~ w, n
!
Hence, finally we have
Now let us supposethat there exists a point sbE Fr(G& ,_ y+-t) with E > 0, such that f admits a holomorphic extension to d(sb, xR) along a channel of width 271R with the Jordan arc joining sb to s, as axis, such that d(s,, nR) = %,o. By definition we have S,($,) = -(y’ + E) < -y+. But as a consequence of the result established above, we must have S,(sb) > --y+. Hence a contradiction establishesthe truth of the theorem. Now we also have COROLLARY 2. If yt = 0 and o* = 0, then Fr(.Y) is a natural boundary for the analytic function dejined by (1) in F’, where F is a connected component of C2*0. If Q*O has more than one connected component, then to each connected component there corresponds an analytic function dejked by (1) in that component.
ACKNOWLEDGMENT The second author wishes to thank the French government for financial support during the preparation of this work. REFERENCES 1. M. BLAMBERT AND M. BERLAND, Sur la convergence des ClCments L-dirichlttiens, C. R. Acad. Sci. Paris 280 (197.5), 263-266. 2. M. BLAMBERT AND R. PARVATHAM, Ultraconvergence et singularitts pour une classe de sbries d’exponentielles, Ann. Inst. Fourier 29 (1979), 239-262. 3. M. BLAMBER~ AND R. PARVATHAM, Sur les ordres infkrieur et supCrieur des fonctions entitres dttinies par des iliments L-dirichlittiens, to appear. 4. M. M. DZRBASJAN, Uniqueness theorems for analytic functions asymptotically representable by Dirichlet-Taylor series, Math. U.S.S.R. Sbornik 20 (1973), 603-649. 5. S. MANDELBROJT. Siries adhkrentes, rkgularisation des suites applications, Paris, 1952.