Let
p
(
ζ
)
p(\zeta )
be a positive function defined on the unit circle
T
\mathbb {T}
and satisfying the condition
|
p
(
ζ
2
)
−
p
(
ζ
1
)
|
≤
c
0
log
e
|
ζ
2
−
ζ
1
|
,
ζ
1
,
ζ
2
∈
T
,
\begin{equation*} |p(\zeta _2)-p(\zeta _1)|\le \frac {c_0}{\log \frac {e} {|\zeta _2-\zeta _1|}}, \quad \zeta _1,\zeta _2\in \mathbb {T}, \end{equation*}
p
−
=
min
ζ
∈
T
p
(
ζ
)
p_-=\min _{\zeta \in \mathbb {T}}p(\zeta )
. Futhermore, let
0
>
α
>
1
0>\alpha >1
,
r
≥
0
r\ge 0
,
r
∈
Z
r\in \mathbb {Z}
, and assume that
p
−
>
1
α
p_->\frac {1}{\alpha }
. Define a class of analytic functions in the unit disk
D
\mathbb {D}
as follows:
f
∈
H
r
+
α
p
(
⋅
)
f\in H^{p(\,\cdot \,)}_{r+\alpha }
if
sup
0
>
ρ
>
1
sup
0
>
|
θ
|
>
π
∫
0
2
π
|
f
(
r
)
(
ρ
e
i
(
λ
+
θ
)
)
−
f
(
r
)
(
ρ
e
i
λ
)
|
θ
|
α
|
p
(
e
i
λ
)
d
λ
>
∞
.
\begin{equation*} \sup _{0>\rho >1}\,\sup _{0>|\theta |>\pi } \int ^{2\pi }_0 \bigg |\frac {f^{(r)}(\rho e^{i(\lambda +\theta )})-f^{(r)}(\rho e^{i\lambda })} {|\theta |^{\alpha }}\bigg |^{p(e^{i\lambda )}}\,d\lambda >\infty . \end{equation*}
The following main results are proved.
Theorem 1. Let
f
∈
H
r
+
α
p
(
⋅
)
,
f\in H^{p(\,\cdot \,)}_{r+\alpha },
and let
I
I
be an inner function,
f
/
I
∈
H
1
f/I\in H^1
. Then
f
/
I
∈
H
r
+
α
p
(
⋅
)
f/I\in H^{p(\,\cdot \,)}_{r+\alpha }
.
Theorem 2. Let
f
∈
H
r
+
α
p
(
⋅
)
,
f\in H^{p(\,\cdot \,)}_{r+\alpha },
and let
I
I
be an inner function,
f
/
I
∈
H
∞
f/I\in H^{\infty }
. Assume that the multiplicity of every zero of
f
f
in
D
\mathbb {D}
is at least
r
+
1
r+1
. Then
f
I
∈
H
r
+
α
p
(
⋅
)
fI\in H^{p(\,\cdot \,)}_{r+\alpha }
.