Suppose that
f
f
is holomorphic in the unit disk
D
\mathbb D
and
f
(
D
)
⊂
D
f(\mathbb D)\subset \mathbb D
,
f
(
0
)
=
0
f(0)=0
. A classical inequality due to Littlewood generalizes the Schwarz lemma and asserts that for every
w
∈
f
(
D
)
w\in f(\mathbb D)
, we have
|
w
|
≤
∏
j
|
z
j
(
w
)
|
|w|\leq \prod _j |z_j(w)|
, where
z
j
(
w
)
z_j(w)
is the sequence of pre-images of
w
w
. We prove a similar inequality by replacing the assumption
f
(
D
)
⊂
D
f(\mathbb D)\subset \mathbb D
with the weaker assumption Diam
f
(
D
)
=
2
f(\mathbb D)=2
. This inequality generalizes a growth bound involving only one pre-image, proven recently by Burckel et al. We also prove growth bounds for holomorphic
f
f
mapping
D
\mathbb D
onto a region having fixed horizontal width. We give a complete characterization of the equality cases. The main tools in the proofs are the Green function and the Steiner symmetrization.