Let
f
f
be an inner function in the unit ball
B
n
⊂
C
n
B_n \subset \mathbb {C}^n
,
n
≥
1
n\ge 1
. Assume that
\[
sup
z
∈
B
n
|
R
f
(
z
)
|
(
1
−
|
z
|
2
)
1
+
β
(
1
−
|
f
(
z
)
|
2
)
2
>
∞
,
\sup _{z\in B_n} \frac {|\mathcal {R} f(z)|(1-|z|^2)^{1+\beta }}{\left (1-|f(z)|^2 \right )^2} > \infty ,
\]
where
β
∈
(
0
,
1
)
\beta \in (0,1)
and
R
f
\mathcal {R} f
is the radial derivative. Then, for all
α
∈
∂
B
1
\alpha \in \partial B_1
, the set
{
ζ
∈
∂
B
n
:
f
∗
(
ζ
)
=
α
}
\{\zeta \in \partial B_n:\, f^*(\zeta ) =\alpha \}
has a non-zero real Hausdorff
t
2
n
−
1
−
β
t^{2n-1-\beta }
-content, and it has a non-zero complex Hausdorff
t
n
−
β
t^{n-\beta }
-content.