Abstract
We consider the space
C
n
Ω
,
the Banach space of continuous functions with
n
derivatives and the
n
th derivative continuous in
Ω
¯
,
where
Ω
⊂
C
is a starlike region with respect to
α
∈
Ω
.
We use the so-called
α
-Duhamel product
f
⊛
α
g
(
z
)
:
=
d
dz
∫
α
z
f
(
z
+
α
-
t
)
g
(
t
)
d
t
=
d
dz
f
∗
α
g
z
to describe usual
∗
α
-generators of the Banach algebra
C
n
Ω
,
∗
α
,
to estimate
I
-
V
α
m
and to estimate below the norm
δ
A
m
,
where
V
α
is the Volterra integration operator defined by
V
α
f
z
=
∫
α
z
f
t
d
t
and
δ
A
is the inner derivation operator defined by
δ
A
X
:
=
X
,
A
.
We give a new proof of Aleman-Korenblum theorem in one particular case. Namely, we describe
V
-invariant subspaces in the Hardy space
H
p
by using Duhamel product.