Let
Δ
\Delta
be the unit disc of
C
\mathbb C
and let
f
,
g
∈
H
o
l
(
Δ
,
Δ
)
f,g \in \mathrm {Hol}(\Delta ,\Delta )
be such that
f
∘
g
=
g
∘
f
f \circ g = g \circ f
. For
A
>
1
A>1
, let
F
i
x
A
(
f
)
:=
{
p
∈
∂
Δ
∣
lim
r
→
1
f
(
r
p
)
=
p
,
lim
r
→
1
|
f
′
(
r
p
)
|
≤
A
}
\mathrm {Fix}_A (f):=\{p \in \partial \Delta \mid \lim _{r \to 1}f(rp)=p, \lim _{r \to 1}|f’(rp)|\leq A \}
. We study the behavior of
g
g
on
F
i
x
A
(
f
)
\mathrm {Fix}_A (f)
. In particular, we prove that
g
(
F
i
x
A
(
f
)
)
⊆
F
i
x
A
(
f
)
g(\mathrm {Fix}_A (f))\subseteq \mathrm {Fix}_A (f)
. As a consequence, besides conditions for
F
i
x
A
(
f
)
∩
F
i
x
A
(
g
)
≠
∅
\mathrm {Fix}_A(f) \cap \mathrm {Fix}_A(g) \neq \emptyset
, we prove a conjecture of C. Cowen in case
f
f
and
g
g
are univalent mappings.