For the contractive iterated function system
S
k
z
=
e
2
π
i
k
/
m
+
ρ
(
z
−
e
2
π
i
k
/
m
)
S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}
with
0
>
ρ
>
1
,
k
=
0
,
⋯
,
m
−
1
0>\rho >1, k=0,\cdots , m-1
, we let
K
⊂
C
K\subset \mathbb {C}
be the attractor, and let
μ
\mu
be a self-similar measure defined by
μ
=
1
m
∑
k
=
0
m
−
1
μ
∘
S
k
−
1
\mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}
. We consider the Cauchy transform
F
F
of
μ
\mu
. It is known that the image of
F
F
at a small neighborhood of the boundary of
K
K
has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of
F
F
away from
K
K
; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.