Let
ϕ
\phi
be a holomorphic function taking the open unit disk
U
U
into itself. We show that the set of nonnegative powers of
ϕ
\phi
is orthogonal in
L
2
(
∂
U
)
L^2(\partial U)
if and only if the Nevanlinna counting function of
ϕ
\phi
,
N
ϕ
N_\phi
, is essentially radial. As a corollary, we obtain that the orthogonality of
{
ϕ
n
:
n
=
0
,
1
,
2
,
…
}
\{\phi ^n: n=0,1,2,\ldots \}
for a univalent
ϕ
\phi
implies
ϕ
(
z
)
=
α
z
\phi (z) = \alpha z
for some constant
α
\alpha
. We also show that if
{
ϕ
n
:
n
=
0
,
1
,
2
,
…
}
\{\phi ^n: n=0,1,2,\ldots \}
is orthogonal, then the closure of
ϕ
(
U
)
\phi (U)
must be a disk.