On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives
For a formal power series the conditions on the Gelfond-Leont'ev derivatives are found, under which the series represents a function, analytic in the disk {z : |z| < R}, 0 < R ≤ +∞.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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irk-123456789-1064482016-09-29T03:02:18Z On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives Sheremeta, M.M. Volokh, O.A. For a formal power series the conditions on the Gelfond-Leont'ev derivatives are found, under which the series represents a function, analytic in the disk {z : |z| < R}, 0 < R ≤ +∞. 2007 Article On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives / M.M. Sheremeta, O.A. Volokh // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 241-252. — Бібліогр.: 3 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106448 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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For a formal power series the conditions on the Gelfond-Leont'ev derivatives are found, under which the series represents a function, analytic in the disk {z : |z| < R}, 0 < R ≤ +∞. |
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Sheremeta, M.M. Volokh, O.A. |
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Sheremeta, M.M. Volokh, O.A. On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives Журнал математической физики, анализа, геометрии |
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Sheremeta, M.M. Volokh, O.A. |
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Sheremeta, M.M. |
| title |
On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives |
| title_short |
On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives |
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On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives |
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On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives |
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On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives |
| title_sort |
on a convergence of formal power series under a special condition on the gelfond-leont'ev derivatives |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/106448 |
| citation_txt |
On a Convergence of Formal Power Series Under a Special Condition on the Gelfond-Leont'ev Derivatives / M.M. Sheremeta, O.A. Volokh // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 241-252. — Бібліогр.: 3 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT sheremetamm onaconvergenceofformalpowerseriesunderaspecialconditiononthegelfondleontevderivatives AT volokhoa onaconvergenceofformalpowerseriesunderaspecialconditiononthegelfondleontevderivatives |
| first_indexed |
2025-07-07T18:30:24Z |
| last_indexed |
2025-07-07T18:30:24Z |
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1837013951868567552 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 2, pp. 241�252
On a Convergence of Formal Power Series Under
a Special Condition on the Gelfond�Leont'ev Derivatives
M.M. Sheremeta and O.A. Volokh
Department of Mechanics and Mathematics, Ivan Franko Lviv National University
1 Universitetska Str., Lviv, 7900, Ukraine
E-mail:tftj@franko.lviv.ua
Received February 12, 2004, revised December 28, 2006
For a formal power series the conditions on the Gelfond-Leont'ev deriva-
tives are found, under which the series represents a function, analytic in the
disk fz : jzj < Rg, 0 < R � +1.
Key words: formal power series, Gelfond-Leont'ev derivatives, analytic
function.
Mathematics Subject Classi�cation 2000: 30D50, 30D99.
1. Introduction
Let (fk)
1
k=0 be an arbitrary sequence of complex numbers. For 0 < R � 1
by A(R) we denote the class of analytic functions
f(z) =
1X
k=0
fkz
k; (1)
in the disk fz : jzj < Rg. The denotement f 2 A(0) means further that either
f 2 A(R) for some R > 0 or the series (1) converges only at the point z = 0, i.e.,
A(0) is a class of formal power series. Clearly, A(R2) � A(R1) for all 0 � R1 �
R2 � 1. We say that f 2 A+(R) if f 2 A(R) and fk > 0 for all k � 0.
For f 2 A(0) and l(z) =
1P
k=0
lkz
k 2 A+(0) the formal power series
Dn
l f(z) =
1X
k=0
lk
lk+n
fk+nz
k (2)
is called [1�2] the Gelfond�Leont'ev derivative of the order n. If l(z) = ez, that
is lk = 1=k!, then Dn
l f(z) = f (n)(z) is a usual derivative of the order n. We can
assume that l0 = 1.
c
M.M. Sheremeta and O.A. Volokh, 2007
M.M. Sheremeta and O.A. Volokh
As in [2], let � be a class of all positive sequences � = (�k) with �1 � 1, and
let �� = f� 2 � : ln�k � ak for every k 2 N and some a 2 [0;+1)}. We say
that f 2 A�(0) if f 2 A(0) and jfkj � �kjf1j for all k � 1. Finally, let N be
a class of increasing sequences (np) of nonnegative integers, n0 = 0.
Studying of conditions on the Gelfond�Leont'ev derivatives, under which series
(1) represents an entire function, was started in [2]. In particular, the following
theorems are proved.
Theorem A. Let (np) 2 N . In order that for every � 2 �, f 2 A(0) and
l 2 A+(1) the condition (8p 2 Z+)fDnp
l f 2 A�(0)g implies f 2 A(1), it is
necessary and su�cient that lim
p!+1
(np+1 � np) <1.
Theorem B. Let (np) 2 N , l 2 A+(1) and the sequence (lk�1lk+1=l
2
k) be
nondecreasing. In order that for every � 2 �� and f 2 A(0) the condition (8p 2
Z+)fDnp
l f 2 A�(0)g implies f 2 A(1), it is necessary and su�cient that
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
�
pX
j=1
ln
1
lnj�nj�1+1
9=
; = +1: (3)
A problem on �nding conditions on l 2 A+(0), � 2 � and (np) 2 N , un-
der which the condition (8p 2 Z+)fDnp
l f 2 A�(0)g implies f 2 A(R), R > 0,
is natural. In [3] the following analog of Th. A is proved.
Theorem C. Let (np) 2 N and let R[f ] and R[l] be the radii of developments
into power series of f and l. The condition lim
p!1
(np+1 � np) < +1 is necessary
and su�cient in order that for every � 2 �, f 2 A(0) and l 2 A+(0) the condition
(8p 2 Z+)fDnp
l f 2 A�(0)g implies the inequality R[f ] � PR[l] with some constant
P > 0.
The main result of this paper is the following analog of Th. B.
Theorem 1. Let (np) 2 N . In order that for every f 2 A(0), l 2 A+(0) and
� 2 � such that the sequence (lk�1lk+1=l
2
k) is nondecreasing and �k�1�k+1=�
2
k � 1,
k � 2, the condition (8p 2 Z+)fDnp
l f 2 A�(0)g implies f 2 A(R), it is necessary
and su�cient that
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; � ln R: (4)
None of the conditions on � 2 � and l 2 A+(0) in Th. 1 can be dropped in
general.
242 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On a Convergence of Formal Power Series...
2. Proof of Theorem 1
In [2] the following lemma is proved.
Lemma 1. If � 2 �, (np) 2 N , f 2 A(0), l 2 A+(0) and D
np
l f 2 A�(0) for
all p 2 Z+ then
jfnp+kj � jf1jlp1lnp+k
�k
lk
pY
j=1
�nj�nj�1+1
lnj�nj�1+1
(5)
for all p 2 Z+ and k = 2; : : : ; np+1 � np + 1.
First we prove the following theorem using Lem. 1.
Theorem 2. Let (np) 2 N and the sequence � 2 � and the function l 2 A+(0)
be such that for all p 2 Z+ and k = 2; : : : ; np+1 � np
ln
lnp+k�1lnp+k+1
l2
np+k
� ln
lk�1lk+1
l2
k
+ ln
�k�1�k+1
�2
k
� 0: (6)
If D
np
l f 2 A�(0) for all p 2 Z+ then the estimate
ln R[f ] � lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; (7)
is true and sharp.
P r o o f. From (5) for p!1 we have
ln jfnp+kj
np + k
� 1
np + k
8<
:ln lnp+k � ln lk + ln �k + p ln l1 +
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
;+ o(1): (8)
We put
Ap = p ln l1 +
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
and
k =
k;p =
1
np + k
fln lnp+k � ln lk + ln �k +Apg; k = 1; 2; : : : ; np+1 � np + 1:
Then
k �
k�1 =
Æk
(np + k)(np + k � 1)
; k = 2; : : : ; np+1 � np + 1; (9)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 243
M.M. Sheremeta and O.A. Volokh
where
Æk = (np + k � 1)(ln lnp+k � ln lk + ln �k)
� (np + k)(ln lnp+k�1 � ln lk�1 + ln �k�1)�Ap:
In view of (6)
Æk+1 � Æk = (np + k)
ln
lnp+k�1lnp+k+1
l2np+k
� ln
lk�1lk+1
l2k
+ ln
�k�1�k+1
�2k
!
� 0;
k = 2; : : : ; np+1 � np;
i.e., Æ2 � � � � � Ænp+1�np+1. If all Æk � 0, then in view of (9)
k �
k�1 for all
k = 2; : : : ; np+1 � np + 1 and maxf
k : 2 � k � np+1 � np + 1g =
np+1�np+1.
If all Æk � 0, then
k �
k�1 for all k = 2; : : : ; np+1 � np + 1 and maxf
k : 2 �
k � np+1 � np + 1g =
1. Finally, if Æ2 � � � � � Æk0�1 < 0 � Æk0 � : : : Ænp+1�np+1
for some k0; 2 � k0 � np+1 � np + 1, then
k0�1 <
k0�2 < � � � <
1 and
k0�1 �
k0 � � � � <
np+1�np+1. Thus,
maxf
k : 1 � k � np+1 � np + 1g = maxf
1;
np+1�np+1g:
Since
1 =
1
np + 1
fln lnp+1 � ln l1 + ln �1 +Apg;
and
np+1�np+1 =
1
np+1 + 1
fln lnp+1+1 � ln lnp+1�np+1 + ln �np+1�np+1 +Apg
=
1
np+1 + 1
fln lnp+1+1 � ln l1 +Ap+1g;
from (8) for 1 � k � np+1 � np + 1 we have
ln jfnp+kj
np + k
� max
�
ln lnp+1 +Ap
np + 1
;
ln lnp+1+1 +Ap+1
np+1 + 1
�
+ o(1); p!1;
i.e., for p!1
1
np + k
ln
1
jfnp+kj
� min
�
1
np + 1
�
1
ln lnp+1
�Ap
�
;
1
np+1 + 1
�
ln
1
lnp+1+1
�Ap+1
��
+ o(1):
Hence it follows
ln R[f ] � lim
p!1
1
np + 1
�
1
ln lnp+1
�Ap
�
;
244 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On a Convergence of Formal Power Series...
that is in view of the de�nition of Ap the estimate (7) is proved.
For the proof of its sharpness we consider a power series
f(z) =
1X
k=0
fnk+1z
nk+1: (10)
Since for the series (10)
D
np
l f(z) =
1X
k=p
lnk�np+1
lnk+1
fnk+1z
nk�np+1;
then D
np
l f 2 A�(0) for all p 2 Z+ if and only if for all p 2 Z+ and k > p
lnk�np+1
lnk+1
jfnk+1j � �nk�np+1
l1
lnp+1
jfnp+1j: (11)
It is easy to see that if f1 > 0 and
fnk+1 = f1l
k�1
1
lnk+1
kY
j=1
�nj�nj�1+1
lnj�nj�1+1
; k � 1; (12)
then (11) holds if and only if for all p 2 Z+ è k > p
kY
j=p+1
�nj�nj�1+1
lnj�nj�1+1
� l
p+1�k
1
�nk�np+1
lnk�np+1
: (13)
We suppose that l1 � 1, and �k=lk = expf(k � 1)'(k � 1)g, k � 2, where ' is
positive, continuous and nondecreasing function on [0; +1). Then
kY
j=p+1
�nj�nj�1+1
lnj�nj�1+1
�
kY
j=p+1
e(nj�nj�1)'(nj�nj�1) �
kY
j=p+1
e(nj�nj�1)'(nk�np)
= e(nk�np)'(nk�np) =
�nk�np+1
lnk�np+1
� l
p+1�k
1
�nk�np+1
lnk�np+1
;
i.e., (13) holds and, thus, D
np
l f 2 A�(0) for all p 2 Z+. Since for the series (10)
with the coe�cients (12) the equality
ln R[f ] = lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; (14)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 245
M.M. Sheremeta and O.A. Volokh
is true, then we need to show that there exist sequences (lk) and (�k) such that
�k=lk = expf(k � 1)'(k � 1)g, k � 2, and the condition (6) holds.
Since for �k=lk = expf(k � 1)'(k � 1)g the condition (6) takes the form
ln
lnp+k�1lnp+k+1
l2
np+k
+ (k � 2)'(k � 2) + k'(k) � 2(k � 1)'(k � 1) � 0;
it is su�cient to choose a sequence (lk) such that lk�1lk+1 � l2k; k � 2, and a func-
tion ' such that the function x'(x) is convex. The proof of Th. 2 is complete.
P r o o f of Theorem 1. At �rst we remark that if � 2 �, l 2 A+(0),
the sequence (lk�1lk+1=l
2
k) is nondecreasing and �k�1�k+1=�
2
k � 1, k � 2, then
the condition (6) of Th. 2 holds. Therefore, if (4) holds, then (7) implies the
inequality R[f ] � R, i.e. f 2 A(R). The su�ciency of (4) is proved.
On the other hand, from the proof of Th. 2 it follows that there exist f 2 A(0),
� 2 �, l 2 A+(0) (for example, lk = 1 and �k = expf(k � 1)'(k � 1)g; k � 2)
such that the sequence (lk�1lk+1=l
2
k) is nondecreasing, �k�1�k+1=�
2
k � 1 for k � 2
and D
np
l f 2 A�(0) for all p 2 Z+ and the equality (14) holds. Therefore, if the
condition (4) does not hold, then for the series (10) with the coe�cients (12) we
have
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; < ln R;
i.e., f 62 A(R). Theorem 1 is proved.
3. Essentiality of the Conditions in Theorems 1�2
We suppose that np = 2p for p � 1 (thus, np+1 � np = np for p � 2) and
consider a power series
f(z) =
1X
k=0
�
fnkz
nk + fnk+1z
nk+1
�
; (15)
where f0 = 0; f1 = 1; fn1 = �n1 ,
fnk = lnk�nk�1
k�2Y
j=0
�nj+1; k � 2; fnk+1 = lnk+1
k�1Y
j=0
�nj+1; k � 1; (16)
and (�n) is an arbitrary sequence of positive numbers. Since for the series (15)
D
np
l f(z) =
1X
k=p
�
lnk�np
lnk
fnkz
nk�np +
lnk�np+1
lnk+1
fnk+1z
nk�np+1
�
;
246 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On a Convergence of Formal Power Series...
then D
np
l f 2 A�(0) if and only if for all k � p+ 1
lnk�np+1
lnk+1
fnk+1 � �nk�np+1
l1
lnp+1
fnp+1;
lnk�np
lnk
fnk � �nk�np
l1
lnp+1
fnp+1:
If l1 = 1 then hence it follows that D
np
l f 2 A�(0) for all p � 0 if and only if for
all p � 1
�np �
�np+1�np
lnp+1�np
=
�np
lnp
(17)
and for all p � 0
k�1Y
j=p
�nj+1 �
�nk�np+1
lnk�np+1
; k � p+ 1; �nk�1
k�2Y
j=p
�nj+1 �
�nk�np
lnk�np
; k � p+ 2: (18)
Choosing properly the sequences (lk), (�k) and (�k), we can show that the
conditions in Ths. 1 and 2 are essential.
For example, if lk = �k and �k = 1 for all k � 1, then the inequalities (17)
and (18) are obvious and D
np
l f 2 A�(0) for all p 2 Z+.
Besides, if l2j = e�2ja; l2j+1 = e�(2j+1)b and b > a, then the condition (6)
does not hold,
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; = b
and
ln R[f ] = lim
p!+1
1
np
ln
1
lnp
= a;
i.e., the inequality (7) does not hold and, thus, the condition (6) in Th. 2 can not
be dropped in general.
Now we show that the condition �k�1�k+1=�
2
k � 1, k � 2, in Th. 1 can not be
dropped in general. For this purpose we put lk = 1 and �k = �k for k � 1, and
we choose the sequence (�k) such that �2j+1 = 1, �2(j+1) � �2j for all j � 1 and
ln�nk = nk, k � 1. Due to the choice l 2 A+(0), the sequence (lk�1lk+1=l
2
k) is
nondecreasing and it is easy to verify the ful�llment of conditions (17) and (18),
i.e., D
np
l f 2 A�(0) for all p 2 Z+. Besides,
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
; = 0;
and
ln R[f ] = lim
p!+1
1
np
ln
1
fnp
= lim
p!+1
1
np
ln
1
�np�1
= �1
2
< 0;
i.e., the condition (4) holds with R = 1, but f 62 A(R).
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 247
M.M. Sheremeta and O.A. Volokh
Finally, we show that the condition of nondecreasing for the sequence
(lk�1lk+1=l
2
k) in Th. 1 can not be dropped in general. We choose �k = ek
2
, l2k =
e�(2k)
2
, l2k+1 = e�12(2k)
2
and �k = 1=lk. Then �k�1�k+1=�
2
k � 1, k � 2, and the
sequence (lk�1lk+1=l
2
k) is not nondecreasing. The inequality (17) is obvious and
for k � p+ 1
k�1X
j=p
ln �nj+1 = �
kX
j=p+1
ln lnj�nj�1+1 = 12
kX
j=p+1
(nj � nj�1)
2 � 12(nk � np)
2
= � ln lnk�np+1 < ln
�nk�np+1
lnk�np+1
;
that is the �rst inequality in (18) holds. Further, for k � p+ 2 we have
ln �nk�1 +
k�2X
j=p
ln �nj+1 = � ln lnk�1 �
k�2X
j=p
ln lnj+1 = n2k�1 + 12
k�2X
j=p
n2j
= 4k�1 + 12
k�2X
j=p
4j = 4k�1 + 4k � 4p+1 < 2(2k � 2p)2 = 2(nk � np)
2 = ln
�nk�np
lnk�np
;
that is the second inequality in (18) holds and, thus, D
np
l f 2 A�(0) for all p 2 Z+.
Besides,
lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
;
= lim
p!+1
1
np
8<
:12n2p �
pX
j=1
((nj � nj�1 + 1)2 + 12(nj � nj�1)
2)
9=
;
= lim
p!+1
1
np
8<
:12n2p � 13
pX
j=1
(nj � nj�1)
2 � 2
pX
j=1
(nj � nj�1)�
pX
j=1
1
9=
;
= lim
p!+1
1
2p
�
12 4p � 13
3
(4p � 1)� 2p+1 � p
�
= +1
and
ln R[f ] = lim
p!+1
1
np
ln
1
fnp
= lim
p!+1
1
np
8<
:ln
1
lnp
� ln
1
lnp�1
�
p�2X
j=0
ln
1
lnj+1
9=
;
248 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On a Convergence of Formal Power Series...
lim
p!+1
1
np
8<
:n2p � n2p�1 � 12
p�2X
j=0
n2j
9=
; = lim
p!+1
1
2p
8<
:4p � 4p�1 � 12
p�2X
j=0
4j
9=
;
= lim
p!+1
1
2p
(�4p�1 + 4) = �1;
that is the condition (4) holds with R = +1, but f 62 A(1).
4. Supplements and Remarks
Here we consider the case when the sequence � 2 � satis�es a condition of
the form � 2 ��.
Proposition 1. Let (np) 2 N , the function l 2 A+(0) be such that the sequence
(lk�1lk+1=l
2
k) is nondecreasing and ln �k � a(k � 1) for all k � 1 and some
a 2 (0; +1). If D
np
l f 2 A�(0) for all p 2 Z+, then the estimate
ln R[f ] � lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
9=
;� a (19)
is true and sharp.
Indeed, from the conditions ln �k � a(k � 1) for all k � 1 and D
np
l f 2 A�(0)
for all p 2 Z+ it follows that D
np
l f 2 A��(0) for all p 2 Z+, where ln ��k =
a(k � 1). It is clear that ��k�1�
�
k+1 = (��k)
2 and, since the sequence (lk�1lk+1=l
2
k)
is nondecreasing, the condition (6) of Th. 2 holds. Therefore, from (7) we obtain
ln R[f ]
� lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
�
pX
j=1
ln ��nj�nj�1+1
9=
;
� lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
� a
pX
j=1
(nj � nj�1)
9=
; ;
whence the inequality (19) follows.
For the proof of sharpness of the inequality (19) it is su�cient to consider
the series (10) with the coe�cients (12) and choose �k = lk = ea(k�1). Then the
inequality (13) holds (thus, D
np
l f 2 A�(0) for all p 2 Z+) and
ln R[f ] = lim
p!+1
1
np + 1
ln
1
fnp+1
= lim
p!+1
1
np + 1
ln
1
lnp+1
= �a
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 249
M.M. Sheremeta and O.A. Volokh
= lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
9=
;� a:
Proposition 1 is proved.
We remark that the condition ln �k � a(k � 1) in Prop. 1 can not be re-
placed in general by the condition ln �k � ak and moreover by the condition
lim
k!1
(ln �n)=n = a. Indeed, let np = p +
�p
p
�
for all p � 0, �k = eak, and
lk = ebk for all k � 2, b > a, and l1 = 1. It is easy to verify that for such �k and
lk the inequality (13) holds. Therefore, for the function (10) with the coe�cients
(12) we have D
np
l f 2 A�(0) for all p 2 Z+. Besides,
ln R[f ] = lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
�nj�nj�1+1
lnj�nj�1+1
9=
;
= lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
9=
;� lim
p!1
a(np + p)
np + 1
=
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
9=
;� 2a;
that is the inequality (19) does not hold.
We remark that from the proof of Prop. 1 it follows that if the sequence
(lk�1lk+1=l
2
k) is nondecreasing, �k = 1 for all k � 1 and D
np
l f 2 A�(0) for all
p 2 Z+, then
ln R[f ] � lim
p!+1
1
np + 1
8<
:ln
1
lnp+1
� p ln l1 �
pX
j=1
ln
1
lnj�nj�1+1
9=
; ; (20)
and moreover the condition �k = 1 can not be replaced in general by the condition
ln �k = o(k), k !1. However the following proposition is true.
Proposition 2. Let (np) 2 N , ln �k = o(k) as k ! 1, l 2 A+(0) and the
sequence (�k�1�k+1=�
2
k) is nondecreasing, where �k = lk=�k. If D
np
l f 2 A�(0)
for all p 2 Z+ then the estimate (20) is true and sharp.
Indeed, from the inequality (5) we have
jfnp+kj � jf1jlp1�np+k
�np+k
�k
pY
j=1
1
�nj�nj�1+1
250 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On a Convergence of Formal Power Series...
for all p 2 Z+ è k = 2; : : : ; np+1 � np + 1, whence in view of the condition
ln �k = o(k), k !1, we have
ln jfnp+kj
np + k
� 1
np + k
8<
:ln �np+k � ln �k + p ln l1 +
pX
j=1
ln
1
�nj�nj�1+1
9=
;+ o(1); p!1:
Since the sequence (�k�1�k+1=�
2
k) is nondecreasing, hence as in the proof of Th. 2
we obtain for all p 2 Z+ and k = 2; : : : ; np+1 � np + 1
1
np + k
ln
1
jfnp+kj
� min
8<
: 1
np + 1
0
@ln
1
�np+1
� p ln l1 �
pX
j=1
ln
1
�nj�nj�1+1
1
A ;
1
np+1 + 1
0
@ln
1
�np+1+1
� (p+ 1) ln l1 �
p+1X
j=1
ln
1
�nj�nj�1+1
1
A
9=
;+ o(1); p!1;
that is
ln R[f ] � lim
p!+1
1
np + 1
8<
:ln
1
�np+1
� p ln l1 �
pX
j=1
ln
1
�nj�nj�1+1
9=
; :
Since �k = lk=�k and ln �k = o(k), k !1, hence we obtain the inequality (20).
For the proof of its sharpness it is su�cient to consider the series (10) with the
coe�cients (12), where �1 = 1, �k = k � 1 and lk = (k � 1)ek�1 for k � 2.
Proposition 2 is proved.
From the proof of Prop. 2 one can see that in Th. À nondecreasing of sequence
(lk�1lk+1=l
2
k) can be replaced by the following condition: there exists a positive
sequence (�k) such that ln �k = O(k); k ! 1, and (�k�1�k+1=�
2
k) does not
decrease, where �k = lk�k.
Finally, the following proposition supplements Th. A.
Proposition 3. For all � 2 � and l 2 A+(0) there exists f 2 A(0) such that
Dn
l f 2 A�(0) for all n � 0 and R[f ] = +1.
Indeed, there exists an increasing to +1 function ' such that
max
�
� 2
k � 1
ln
1
lk�1
; �1
k
ln
�1�k
lk
�
� '(k); k � 1:
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 251
M.M. Sheremeta and O.A. Volokh
We put fk = lk expf�(k + 1)'(k + 1)g; k � 1. Then
1
k
ln
1
fk
� 1
k
ln
1
lk
+ '(k + 1)! +1; k !1;
and for all n � 0 and k � 1
fk+n
lk+n
= e�(k+n+1)'(k+n+1) � e�k'(k)e�(n+1)'(n+1) � l1�k
lk
fn+1
ln+1
;
that is R[f ] = +1 and Dn
l f 2 A�(0) for all n � 0. Proposition 3 is proved.
We remark that in view of Th. A one can not replace R[f ] = +1 by R[f ] =
R 2 (0; +1) in the last proposition.
References
[1] A.O. Gelfond and A.F. Leont'ev, On a Generalisation of Fourier Series. � Mat. Sb.
29 (1951), No. 3, 477�500. (Russian)
[2] M.M. Sheremeta, On Power Series with Gelfond�Leont'ev Derivatives Satisfying a
Special Condition. � Mat. phys., anal., geom. 3 (1996), 423�445. (Russian)
[3] O.A. Volokh and M.M. Sheremeta, On Formal Power Series with Gelfond�Leont'ev
Derivatives Satisfying a Special Condition. � Mat. Stud. 22 (2004), No. 1, 87�93.
(Ukraine)
252 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
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