Let
\[
(
T
a
)
(
y
)
=
∑
n
=
0
∞
(
−
y
)
n
g
(
n
)
(
y
)
n
!
a
n
,
y
≥
0
\left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0
\]
be the sequence-to-function Hausdorff transformation generated by the completely monotone function
g
g
or, what is equivalent, the Laplace transform of a finite positive measure
σ
\sigma
on
[
0
,
∞
)
[0,\infty )
. It is shown that for
1
≤
p
≤
∞
1 \leq p \leq \infty
,
T
T
is a bounded transformation of
l
p
{l^p}
with weight
Γ
(
n
+
s
+
1
)
/
n
!
\Gamma \left ( {n + s + 1} \right ) / n!
into
L
p
[
0
,
∞
)
{L^p}[0,\infty )
with weight
y
s
,
s
>
−
1
{y^s},s > - 1
, whose norm
‖
T
‖
=
∫
0
∞
t
−
(
1
+
s
)
/
p
d
σ
(
t
)
=
C
(
p
,
s
)
\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )
if and only if
C
(
p
,
s
)
>
∞
C\left ( {p,s} \right ) > \infty
, and that for
1
>
p
>
∞
,
‖
T
a
‖
p
,
s
>
C
(
p
,
s
)
‖
a
‖
p
,
s
1 > p > \infty ,{\left \| {Ta} \right \|_{p,s}} > C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}
unless
a
n
{a_n}
is a null sequence. Furthermore, if
1
>
p
>
r
>
∞
,
0
>
λ
>
1
1 > p > r > \infty ,\,\;0 > \lambda > 1
and
σ
\sigma
is absolutely continuous with derivatives
ψ
\psi
such that the function
ψ
r
(
t
)
=
t
−
1
/
r
ψ
(
t
)
{\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )
belongs to
L
1
/
λ
[
0
,
∞
)
{L^{1 / \lambda }}[0,\infty )
, then the transformation
(
T
λ
a
)
(
y
)
=
y
1
−
λ
(
T
a
)
(
y
)
\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )
is bounded from
l
p
{l^p}
to
L
r
[
0
,
∞
)
{L^r}[0,\infty )
and has norm
‖
T
λ
‖
≤
‖
ψ
r
‖
1
/
λ
\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}
. The transformation
T
T
includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on
l
p
{l^p}
which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.