THE RICE INSTITUTE
ON FUNCTIONS HOLOMORPHIC IN A STRIP
AND
MANDELBROJT* S INEQUALITY
Hans Jakob Reiter
A THESIS
SUBMITTED TO THE FACULTY
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS
Houston, Texas
May I960
DEM ANDENKEN MEINER ELTERN
INTRODUCTION
1.
This paper is concerned with topics, developed
by S. Mandelbrojt
( /4/,/6/) ^ in the
asymptotic series. For a general survey
theory
of
of the ideas,
see his address /?/ .
The first of these papers contains the
that the familiar inequalities of Cauchy,
discovery
for
the
coefficients of Tayloris series, may be extended,
a very general way, to asymptotic series
in
representing
a holomorphic function in an infinite rectangular strip.
In the second paper, the theory is further developed
so as to cover more general strip regions,
with
a
restriction, however, concerning the width of the region
at infinity.
An important step in that development is the solution
of a problem which is an- extension of the well known
problem of Watson. This was carried:uout in a joint paper
by Mandelbrojt and G.R. MacLane /5/» for strips
having
a finite width, different from zero, at the point infinity.
(l) Numbers in brackets refereto the references listed at
the end of the paper.
-ii-
2. It is the object of this paper
to frame the
proofs in such a way that this restriction,
resulting restriction on Mandelhrojt1s
and
the
inequality,
may he removed. This is done by a suitable modification
of the other conditions, assuming, e.g., existence
the derivative, instead of bounded variation,
function representing the boundary
of
the
of
of the
strip,
together with a certain convergence condition.
The main tools used in the solution of
the
generalized problem of Watson, are Ahlfors' inequalities
for the conformal mapping of infinite strips.
of the second inequality,
here,
for the regions
A proof
considered
is given.
3. I express my thanks to Professor S. Mandelbrojt,
for proposing the problem, and the lively
and to Professors
discussions,
B.L. van der Waerden and Andr4
for their encouraging interest.
I
Weil,
1.
AHLFORS* INEQUALITIES.
Let
As
be a domain in the
defined by
t
S-plane
( j * r + ii )
), 1*14 &-r>
, where the
function is continuous and > 0 for <r> c .
Denote by
maps
A$
Z -plane
2 = Z (■$) = x(s)-\r "$(s) the function which
conformally upon the strip J[ • U( ^ - in the
( 2 « x-t >ij
a) the interval
( — oo < y < oo
)
) in the following \vay:
( c , oo )
of the
is mapped on the real axis
z -plane
b) for some chosen value of <r , say
x
*
, we have
0
c) z*(<r) > 0 for <r> c.
The function
Denote by
z * z(*)
is uniquely determined.
5 ~ -£/z) =■ vy*; t i\(t)
its inverse function.
It should be remarked also that the mapping is continuous
and one-to-one on the boundary.
Set
x (G) =
'Vy'ii-h x (<) * 'Witfix
ltj 6
)tl &
£ (-*) «
'Vvn / X -i-15
Ijl*{
Ahlfors* first inequality
x(<r+U)
<3(4 f
(see /l/ or /8/ ), ^
(1) Numbers in brackets refer to the references listed at
the end of the paper.
2-
-
applied to our domains, states:
If
(<rz ? <T, /► C)
are such that
j-22—
c
(A^
then
x(«i)
- 3C(r.,
>
zl-0-
-
Lr)
p, ^
i
V.
*1
It follows from
(A^) that
<f(*)
(1.1)
J <?<■%
S 8
£(*)
A proof of Ahlfors’ second inequality will now be given;
the explicit statement of that inequality will be deferred
till the end of the proof
In the strip
abscissa
x
;
fig. 1, p.Zf).
(p * J
).
consider the vertical segment of
denote by
Let
TV*)
j
its map in
(see
be the area of that part of A
,
which is contained between a fixed segment of abscissa
and the segment of abscissa 6 ,
number which is such that the area
Denote now by
A. ( «(*))
(*>)
the
is equal to
the area of that part of
contained between the fixed
segment of abscissa
and the arc
rw.
Let 'hi(<r,)be the greatest lower bound of the lengths of
-3all Jordan arcs which connect the upper and lower boundaries
of
and meet the segment of abscissa .
The segment of abscissa
r~r
c* (x)
cuts the arc J (x) ;
thus
S
_T
J
and, by Schwarz1s Inequality,
/
MW
<
Zj
_/
Now
A
%\
4 A («M)
_
Jl
TJ
«
£
(<?/ =
<
» and therefore
T•
X'C*) ’
4K
4«
Thus we have
Ax
C*
<?<*>
<
ZT
'hi (<*)
4*(
'
O<(A)J
and finally, on integrating,
\1 - x,
<•
Zr I ~}°J
J
.
d<r
.
°<l*i)
Let us now set
^ » 3~L<rK)
*(■<■,) -*(*■,)
,
S Zir j
X, - * Cc,)
3l<r)
>
, tic .
then
(1)
J Mfirj*
The integral on the right may be split into three parts:
(1) The preceding reasoning is due to Ferrand and Dufresnoy
/3/ j cf. also a forthcoming book by S. Mandelbrojt.
-4-
«(*«<))
+
*/
«( * Ccr,))
t
1
I
*
Indeed we have (cf. fig. 2, p,13):
£Ut<r()) $
r
*
of(xt<r,))
£ ?(x^j
*
, <f,
?
^
FOFtnrt .
<
We may now write
I
*C6.t<r>)) *t*C*i))
GJ
‘
‘
£(6.t<rt)) <rt
£(*«>))
At this point, we need some information about the
asymptotic behavior of This behavior depends, of
course, on the properties of the function , and
for that reason it will be convenient to impose the fol¬
lowing restriction on
i)
exists,
:
—y oo( s' —oo )
Consider any rectifiable Jordan arc connecting the
upper and lower boundaries of and meeting the
segment of abscissa
<5"
in some point
JM
;
the
length of this arc is not less than the sum of the shortest
distances from to the upper and lower boundaries of
, respectively.
Let now
Htc,±)
be a point on
-5
the segment such that this sum is a minimum, and let
fj. («*,
f
gCffi),)
est points
be
»
(see fig. 3, p.
the
corresponding near¬
).
We have
> ^(<r; > gf<r,;
How
| <3*^ - <s~ j
'M*-)
i l3L<r)
f
l *cannot exceed
*- .
2 o(v)
t since
.
We shall now prove:
Lemma 3.: If o' —> oo , cr1 —& t*
]<T-<5',| <
where
£(<H )
k
in such a way that
is constant, then
£t<r)
^
(<r —* •*) .
Proof: This follows immediately from the mean value
theorem.
We have
I %L«') - £&)] *
where
6s *
is between
l<r' -<rhl $'(<?-*)) j
<T
‘
and
1
I
<y1 .
Thus
- k I S'M
6-
-
The term on the right tends to zero, as (T —, since
6~1 —> c£
C —s> implies
G~ • —y> o°
, (by condition i)) and hence
•
We have now the following result:
Lemma 2:
(<r) ~ 3.£(«~)
( <r )
Continuing our discussion from p. 4, we may now write,
on combining lemma
2
and the relation
(1.1)
f(*l
[ &L i<r
J %(<r
• 0(i)
Lx -. ») .
£CK)
As a first result, we have now obtained the inequality
- *(«•,)
+ (9W *’ )
< -2T Jr
To investigate the integral on the right, let us write
We want to find an estimate for the second term on the right*
By
(1*2)
we have
JL
<
yfe)
_
l£(r,>+fCVjX
j_
V'
^ (i. il<r)Lgti) +p*,)f *tpr)
How we want to find an estimate of the differences
| <j(«■) - gOr,) |
;
l .1,1.
In order to obtain a sufficiently sharp estimate, we
proceed as follows
The segment
(consider again fig. 3, p.
)*
li. is orthogonal to the curve £ * ^.(p)
and we have
+
(s-<r)
whence
( JK) 't )■ gV,;
*°
\ <r, - <r | = | jV,)]. I
Therefore
j .
l<r, 4
In the same way
^ - cr|
4
+
£<jr)J
We have
|^) - $(«-)) where
<j~* is between
l
I 8-^0 ■"
I
-
l<r, -<r|.
<f"
and
<5*
x
I y't*7) J
Hence
J' I
or
J*-51) ~ I j'^dlTl
-
g(<T)|
f4
-8-
and thus finally
| rfrj
- .ro/
S
i
-
-
M
h'W-s'ft')!
Crlf'/fy. s'((x')]-$<*) /
for <r large*
How we want to Impose a further restriction on the
function
replace
^.(r) t which will permit us,
| ^(st) -fa')! */">•
In effect, to
One very simple and useful restriction is
ii)
S
*
non-increasing.
If this restriction is assumed, the proof can be com¬
pleted In a very simple ways since
- <r/ £
fl s,}4)
we may write
U'l'i)-3'<*i‘)}
£
S'(r ~ *g&f-
Then we have
JM
to (<r) *
n''i s'fc-Zsfc))*
Q
sHisiMf
jf<r - tyc*))
-9-
by lemma
1 .
Thus
Cf—
W
)**- ±
—
r
V ’* * .V"'
,
(yr
<r, S(T-%!<■»
V,
since
o
}i*)
f
f
$(e-Zj/r))
, or
, —
)</<r
.W
T -#Yr)4
< i'*
.
4
" ^ j-jTtr '<v"
Thus finally
rfK£lz
_/.
1
)y,
*
cfc&lr.
J . itr)
c
$
We have now proved Ahlfora1 second inequality
/\f
for the strips considered here, i.e. with g(r)
satisfying the conditions
i) ^C6~) exists, g !Cs)~* O ( —> o° )
ii) j^lC) |
is
non-increasing.
/
tP ,
—
,
«#<r 4. oo
<V
then
—
x
- SYJ
£
/>** i<r
f f
* J
$C<r)
t
-10-
The condition ii) may be replaced by
ii'>
/
'*
- :'(*■>! £ —£
'
/s-'-s-/
7
*
for some constant
j/ -C I
•
This restriction, a kind of a modified Lipschitz condi¬
tion, is rather weak in the case where gfr)
more stringent otherwise.
C<r—>n)Jb\xt
The proof may then be given as
follows: we have
hH)-yt'j/ & —£■—ir -o-i (i.»)
J(r, > +£<*>
!
JL
tgCr)l .
£ (ft) *$M
Thus
h'w -i'M-h rUlroh
Writing
we get
whence, since
//<V/ < ig>(r)i
+■ I $'lV - S'MI
I s'M ± Is'^l * rls'(*i)l
/■•*/
•
I 3'C<r)l
(i* W.
11-
-
In the same way,
If'fy’) -
/M
£
2L
-
JL
<r/
/<r, - S'/
i^i) i- £«V
*
for
<r
large, by lemma
/<?%;;/
^
Hence we have, for
W
1 , and finally
f
<$~
-jV-JJ/ £
large,
!}k) ~ §(si>l
In this way, we have again the desired estimate:
v
t
(
h
SM
—— £ C-f ~3 (<r) *l<r •
J $(*)
Remark: A sharper inequality, with less restric¬
tive assumptions (and a more intricate proof), was given
by Warschawski
If
J(<r)
/10/ :
satisfies condition
i)
only, then
(Actually, Warschawski’s Result is still more general)
On combining Ahlfors’ two inequalities, we obtain
<51
if
1
-
4-r < x(*i)
-
K,
*(«-,) < j
L
J JCf>
1
5
whence it follows, on setting
<rz » <r/ oj * const. :
Lemma 3:
If
r** / z
/ —o\r
J
jW
then
'*•(<>') ~
<
,
'S (<?)
f
0 (l)
(<T-> 00)
Swlfjk ■
<r
where
A sufficient condition for the convergence of
.to
$0
Jr 4
may be obtained by a suitable modification of Hadamard’s
lemma on derivatives (ef. /2/ , p, 12), namely:
■£(*)
If
0
9
is defined for
an<^
x >. xc
, if
{(*)>€> »
non :i ncreas
" -
ft*)1 £ 1 f(*) f "(*■)
inS» then
X > A.
-13Proof: By Taylor’s formula , if
8
{(*+*)
- Q.-f'fcO
-
f (*)
*
f* W 5
+
<*• fU*)
- f(x+a.)
fc*)
£c*>
* >0/
+
£*..
z
.
-t- a•
ot
to** i"t>) > o.
The minimum of the expression on the right >
for
<t > 0 , is
, q.e.d.
The following lemma is now obvious:
Lemma 4: If the function is such that
j C<rV
/<f)i o ,
exists and is non-increasing, then
I
Remark: If g(<r)
j'tf </<r < oo
jtr>
satisfies condition
i) and ii), and j(c) ck <
^(f) > ° , the convergence of the integral is obvious.
We are now going to prove the following result, which
will be needed later on in the paper:
Lemma 5:
Aj.tr} * _jr
et<r
<2 3(0
C 6- —> co )
The proof is a straightforward application of Ahlfors’
inequalities and of the following result due to Ostrowski:
14If
X
,
X, —i> 0O in such a way that (x - x, 1 < H »
a constant, then
<r f
o'* (x)
' (*,) ~
(x -*<P) ,
For a proof of this result, and references, the
reader is referred to Warschawski’s paper /10/, theorem 2,
(b) .
It will be convenient to divide the proof of lemma
5
into several parts:
Lemma 6: Let
be an arbitrary constant.
Then
(
;
4
)
Proof: This is an immediate consequence of the mean value
theorem for integrals, and lemma
Lemma
«■. >
If <T
»
1 ,
* , <r^ —in such a way that
<rj — G“ < Pl gCr) IM
a constant), and if
«/<r < oo
3(t)
then
/'((f,)
x' (.<r)
~
( <r -*• «®) .
Proof: From Ahlfors’ second inequality, and from lemma
it follows that
0 4
x(C,)
-
x (O
<
0 (}) ( 6<fi) «
Using now Ostrovski’s result stated above, we have
x1 ( et)
Y' (<r)
_
6,
(<)
)
-15( € —pv* implies pc —p oo)
, and the proof is complete.
Now we have all the tools for the proof of lemma 5.
For an arbitrary positive constant
3*
consider the dif¬
ference
- xfr) - *'£(*)■• x’Ct'Oy
*(%'£(<?))
<S” ^ (S’ • 6" +• $ ■ji*)
where
From Ahlfors’ inequali-
ties we have
G+tL-g(f)
~*r-
tJgM
c
> s^) ^(r‘) ~l
*
Since the derivative of
r
*
*'[*')LZ
y
and letting
$(ri
s
W
v£f{!:J-£r+K
(f) lk.3 sw
, we obtain, by lemmas
- *nrj <
4'
6
and
7,
jCr)-x!t<r) < j -1 f'* + ft]j
<r—p oo
where K
K•
is positive, we may write
,
G+a'jg(rJ
'"J ~
g- —> oo
h [j‘4
t (f)
ter +
r
is independent of CL
..
Since
&,
-
-*> '
may be taken
arbitrarily large, we obtain the result stated in lemma
5.
The reader is again referred to Warschawskifs paper /10/,
pp. 318, 319, where the result is proved with less restric¬
tive assumptions on the strip (cf. p. 11), and without
supposing
f Jf
. OQ
.
-162.
THE GENERALIZED THEOREM OF MANDELBROJT AND MACLANE.
The theorem of Mandelbrojt and MacLane (theorems I
and II in
/5/ ) is an extension of a lemma of which a
simple proof was given by Ostrowski
/2/ , note I ).
This lemma
/%/ (cf. also Carleman
(theorem
A*
in
/5/ ), which
plays an important role in the solution of Watson's problem,
is as follows:
Let
f?1%)
be holomorphic and bounded in the strip
continuous on the boundary, and not identically
zero.
Then the integrals
are finite.
The statement of the theorem to be proved is as follows;
Theorem
Let A$
<T
be a domain defined by
>& ,
IH i g(r)
where the function is continuous and > O
and satisfies the following conditions:
11
ii)
i' [<fj
exists,
^'( f ) —> 0 (<r —3*
j o^<r)| is non-increasing ^
for
C ,
-17-
r
$(*') 1- $(<r)
for some constant
lii)
/ ^ ^^
F*1 Cs)
Let
^ -C / •
» not identically zero, be holoaiorphic
and bounded in
, continuous in A4
As
the closure of
,
( A4
denotes
, excepting the point at infinity)•
Let N(c) be an increasing function such that for
C'
large
%/ F(.*~+igC*)) < -A/M*
Then the integral
JV> i(e:Scr,J
(SM*
f/±)
is finite.
Proofs
Let us set
£(z)
-
F^CsCz))
, where
J(z)
mapping function defined at the beginning of
is the
section
1 ,
Then we have
%jl§Lx+lf)l ~ &3I F(f(Ki-:i)(r(*+ij)))l £ - j/(c(xr ij)J.
From theorem A*
it follows that
J /\A(^(xrif)J £* <tx
We want to replace by
C
<#
<s"^x")
For that
-18-
purpose, we shall obtain the following estimate:
S' (K)
— CT 6c) <
large.
for
)& g (<r£x)) y
Indeed, recalling relation
(1.1), p. 2,
TC<)
f£(*) — < 3 >
J S(r)
)
we have, by the mean value theorem,
— <r c*)
where
and
£(x.)
<
8
C*)
6~« , since | <r# - <r|
(j>(c{x))
-
£
•
j
Applying lemma
W(x) - £ &)
we
>
1
to
<r*
Set
fx —>oo)» which gives the desired es¬
timate.
Now we can write
Woo
J
- /6 ^f<rCx))) <?
Putting
x « x(<*~)
I ,/Vy<r -
and using lemmas
<fL<r)) £
—-—•
3j(<rj
4
3
oo
and
4<r
,
5 , we get
<C
<x>
,
Now
£(*'))
by the argument of lemma
(C —p ec ^
by lemma
6 , and
1 .
Hence
*
t'fjfiT
^<r)
=
f<T
f ~ ItgC?))
-> (P)
-19-
N [s - f
&
St* - H>gGD)
Tr
■2 £(<r- f6$6r))
or
d<r
c°
4
*
N(<r-H>^(<n) e ^ ^
(f
-
~
tGg’M) c)<5~ <
Z £ (<r i.e.
;tP
MM
-S(T:)
1
'
_
//
«/r 4. CO Q.E.D.
}
Z $(T)
Remark: Condition ii) or ii’)> imposed on the function
, is of course superfluous, if recourse is had
to Warschawski’ s results
/ltt/' *
We shall now prove a converse to theorem
Theorem 2: Let A5
Let
f^(^)
be
an
1 :
"be the domain defined in theorem
1 .
increasing function such that the integral
S^(r)d(e-kr,J
is finite.
-
Then there exists a function
^
J- (£)
holomorphic in As
continuous and not taking the value zero in
I
. holds in
*" £ i y(r))j<r
, where
M (
>
,
such that
tkj M(<r) < — ^C^)
<r)
=
^c^'jF(<rti ir) j.
It I *£(<)
-20-
Proof: We have
P oo
J Ntf
A
or, setting
<r = <TC
-20?) —
a
)
»
d<r
—"—
by lemma
3 ;
^ using lemma
constant
oo
.
(x
o*);
thus
6
X
S N(cr(x * <0,7 e. *
d
<
5 ,
0(0
~
y
and hence
oo
t-st?)
an<
ojx
Now 3 ( <3-(K-))
<.
e{x
<
oo
x <
,0
, for any real
.
Now we can apply lemma
VI
in Mandelbrojt and MacLane
/5/ : there exists a function $ (2.)
holomorphic in
5 , continuous and not taking the value zero in
A2
, such that in
Ax
The function b (s) = %■(&))
is holomorphic in J\s
continuous and not zero in
, and satisfies in A,
£ *
N(clx(<r+Hr) tel]).
Prom Ahlfors’ second inequality it follovs (by setting
C, - <S~z - <S~ )
x [<T * vi)
then
+
that for fli sufficiently large,
d > x(cr)
<f)x.(<n.it)
+
, for all
<r + It
dj } <r{x(<f)):6-, and
in
As
,
-21
which completes the proof.
Remark: If we apply theorem
^
^
2
to the domain
9
|i) <
° j
f
setting
* ife
and noting that
-5
- JS*{c)
convergence of
OfiJ
-
so that the
•
JH(r)clfe'SM)
is equivalent to the convergence of
-
5 \<r)
Al(c) 4 {*-*"-)
and that
( V Cr>
$e have a stronger conclusion to theorem
i* (&)
A
A
2 , namely that
is holomorphic and never zero in the closed strip
(excepting, of course, the point at
satisfies
p*
) and
-22-
3.
MANDELBROJT'S INEQUALITY.
We shall follow the exposition in Maridelbrolt1s
paper
/6/ , with some simplifications.
For the definition of the upper mean density of a
sequence
j
04 ^f •
of the functions
associated sequence
, and for the definition
L'„
»
j .A.**}-ha)
»
, and of the
131:16
reader
ls
referred
to pp. 352 - 354 of the paper above.
We say that a function
domain
As
p^(\s) , holomorphic in a
, is represented there by the sums
vn
'Vw > 'VN
k*\
with logarithmic precision
if, for
3 6 As >
and
*
73* (<r) (
t
00
sufficiently large,
-Wi
/6vujo I
)
-TO* or)
p £s)
^ «r>*
(cf.
/6/ , p. 356).
Theorem (Mandelbro.1t1 s Inequality):
Assumptions:
I) The sequence
density
D*<«-
j
0 4.
»
1:1618
an upper mean
is a domain in the
5 -plane
23-
( £ s. oi-t ) defined by
A5
:
G > c ,
\il
~
4
,
where the function has the following properties:
Si>
$(«■> is continuous, S D-
( <r> c
®2^ exists, jV<r)—^0( C“ —b°
)
), and
||7<0/ is non-increasing.
II)
p1 (s) is a function holomorphic in
, which can
be continued analytically, through a channel of width
larger than
Zn D*
, into the circle
III) The sequence
j°l
J
anc
*,?[
^
integer are
such that the sums
1v>
I 4,
.^
6.
'Vn >
k»i
represent
Pcs)
in t*10 domain with a loga¬
rithmic precision 'jo* (<f)
such that the following condi¬
tions are satisfied:
There exist functions
V
,
C
(<T)
satisfying
h(f) is continuous and non-increasing, and
\\ (c) ^
ho)
h (<?)
D ’ ( <r
oo)
hf(c) exists, l<}lft)-h'(e)l
£'(*)
“
W
—> O
^ fjC6') j
is non-increasing, and
f <r—* ■**)
24-
-
j CjW'-h'M]
}
C1)
elr
<
00
C(<r) f
op
- p^OO
^>5.LK
c
p
<
- 'p>t(<r-2o<)
for some constant <K > ir D* ,
<
~ CO?)
} 'Jf
- £6r>,
2'Cc) >
£(e)
<0
and the following relation holds:
(P
f Cc*) J ( -SO?))
cfu.
5 v) - if
Z J g(k)-h(t,)
where
s=> —
OO
Conclusion:
I
I
Proof:
A ^ (s)
<
E.S
& /l*» R.
ivlv
c
Let us write , for
ts-
VC-i)
-A\
/
Vy» £ h
—
6%. X
k=,
seAj
In the case where »
/SK&
’
<r > c, ^
Jfcj <
TJ"*)
-
O
V\[<r + <xj~\^
-25
(see fig. 4” , p.
at a distance
A5
$0 ).
>
Every point of that strip is
f fa (<? +■ <*)
, vrtiere the function
from the boundary of
/\
fs)
is holomorphic.
The reader is now referred to lemma
brojt’s paper
/6/ , p. 361.
the circle with center
5
For
6 .Aj
^"/
if
=• <?' + i t'
jA
Let
fs)
©<
A.
TT
$
.
, we have
and by condition h^).
is holomorphic in
00
r
=
X
and radius
£ JL O7 - c*-;
(£) I
is large, since
The function
in Mandel-
(1 Cs', T U)
Denote by
S £ (f(s'JT}\(G’/+Of)J
>
I
II
L.
(u)
CVV
/W
^ ^ A <fs;
(for the notation, see
/6/ , p. 361).
'*«)) ■
^
By lemma
^
II
(■A^n£s)| - *Tr h C f1 + <* ) L* (irK (e-' + c*))
But (see
^> M.
/6/ , p, 362, bottom lines)
•A.*,,, - T\ (4) - Z clr eT^AUif) *
r>--' *
Hence is in<^ePendent of
Let, then,
5y
-
5
G
A * Cs) - B^ Ci)
A~
5
■»
we
have
$
/
Vn
i- eS
A*
, for <>n>'h •
4vi > 4o
Taking now
-26-
(BhC$)l
£
T/I
(<T +<X)L (wh(tr-bit))
t<T-oQ
h
—
k • L h (T b
<
k- /-* (xh L<ri-<x))
the function
'W\
2,-h
<x)J ]P^ (•r—**)
&
k- e_^<r<r^°';
B^fe) being holomorphic in
From assumption
III
of the theorem, we have
f<fi -1 f
J C(<n-°<)
J-c
I
e
«/<r = oo
*2 fff<r -t-o<) -fa (<rW
or
/d>
C' <r+<x)'
(U.
-if
<3.
<J/(T
z[^(c-b<x) -i» r<ry-WJ
Hence, by theorem
1 , the function^? ($) must be
identically zero, i.e.
4 G, ”/7*$
F a) * c-o * —-—
A,
This proves that the series
,kjU
1 2k)
(
Lc-<;
c*
r
cs)
k »o rr-_
f" /.
^ (*») 7C 7
converges uniformly in
i1*
Hr
4*
A;
<2
, and is equal there to
»
-27It is clear, according to lemma
that the sum of this series,
/~j
II
( /6/ , p. 361),
(s)
, may be continued
analytically along the central line of any channel of
width larger than
2rD'
, through which the function
itself may be continued analytically.
shows, by assumption
II
A*.h g.* d.)
of the theorem, that
('I) *
But, according to lemma
lf\ (&>)! S
where
This
TR
II
(TR
5
‘
( /6/ , p, 361)
) ■ M(S'twR.) ,
Mfs^irR)? (s & To., 7T R)Hence
r
M(s
0 vR)A Q.E.D.
H
^
If ^f(<r) > o , we consider instead the strip
^5
•• «■>«•/ '
(see fig. 4"^ p. 3o
),
y
K1
4
T
T~
04
) ~
The proof is then analogous.
28-
-
FIG U E ET
I
UPPCE WPLT UPPCB
UQLF
or AZ
or AS
UPPER
UQLF UPPER WOLF
or A &
or A2
FIGURE
a
29-
-
FJG.3
(4+)
FIGUIZ-E: 4-
30-
REFERENCES
I.. L. Ahlfors, Untersuchungen zur Theorie der konformen
Abbildung und der ganzen Funktionen, Acta Societatis
Scientiarum Fennicae (Nova Series), vol. 1 (1930).
.
2 T. Oarleman, Les fonctions quasi-analytiques, Paris,
Gauthier-Villars, 1926.
3. J. Ferrand et J. Dufresnoy, Extension d'une in&galit&
de M.“"Ahlfors et application au probl&me de la d&riv&e
angulaire, Bulletin des Sciences Mathfimatiques, (2) vol. 69
(1945) pp. 165-174.
4. S. Mandelbro.it, Quasi-analyticity and analytic con¬
tinuation - a general principle, Trans, Amer. Math. Soc.
vol. 55 (1944) pp. 96-131.
5. ' S_. Mandelbro.1t and G. R. MacLane, On functions holomorphic in a strip region, and an extension of Watson's prob¬
lem, Trans, Amer, Math, Soc. vol. 61 (1947) pp. 457-467.
6. S. Mandelbro.1t, Sur une infcgalitfe fondamentale, Ann.
Ecole Norm.(3) vol. 63 (1946) pp. 351-378.
7. S, Mandelbrojt, Analytic continuation and infinitely
differentiable functions, Bull. Amer. Math. Soc. vol. 54
(1948) pp. 239-248.
8. R. Nevanlinna, Eindeutige analytische Funktionen,
Berlin, Springer, 1936.
9. A. Ostrowski, Quasianalytische Funktionen und Bestimmtheit
asymptotischer Entwicklungen, Acta Math. vol. 53 (1929) pp.
181-244.
10.
S. E. Warschawski, On c
strips, Trans. Amer. Math, Soc. vol 51 (1942) pp. 280-335.