A Schwarz function on an open domain
Ω
\Omega
is a holomorphic function satisfying
S
(
ζ
)
=
ζ
¯
S(\zeta )=\overline {\zeta }
on
Γ
\Gamma
, which is part of the boundary of
Ω
\Omega
. Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if
Ω
\Omega
is simply connected and
Γ
=
∂
Ω
∩
D
(
ζ
,
r
)
\Gamma =\partial \Omega \cap D(\zeta ,r)
, then
Γ
\Gamma
has to be regular real analytic. This paper is an attempt to describe
Γ
\Gamma
when the boundary condition is slightly relaxed. In particular, three different scenarios over a simply connected domain
Ω
\Omega
are treated: when
f
1
(
ζ
)
=
ζ
¯
f
2
(
ζ
)
f_1(\zeta )=\overline {\zeta }f_2(\zeta )
on
Γ
\Gamma
with
f
1
,
f
2
f_1,f_2
holomorphic and continuous up to the boundary, when
U
/
V
\mathcal {U}/\mathcal {V}
equals certain real analytic function on
Γ
\Gamma
with
U
,
V
\mathcal {U},\mathcal {V}
positive and harmonic on
Ω
\Omega
and vanishing on
Γ
\Gamma
, and when
S
(
ζ
)
=
Φ
(
ζ
,
ζ
¯
)
S(\zeta )=\Phi (\zeta ,\overline {\zeta })
on
Γ
\Gamma
with
Φ
\Phi
a holomorphic function of two variables. It turns out that the boundary piece
Γ
\Gamma
can be, respectively, anything from
C
∞
C^\infty
to merely
C
1
C^1
, regular except finitely many points, or regular except for a measure zero set.