Let D be a bounded homogeneous domain in
C
n
{\mathbb {C}^n}
, and let
Δ
\Delta
denote the open unit disk. If
z
∈
D
z \in D
and
f
:
D
→
Δ
f:D \to \Delta
is holomorphic, then
β
f
(
z
)
{\beta _f}(z)
is defined as the maximum ratio
|
∇
z
(
f
)
x
|
/
H
z
(
x
,
x
¯
)
1
/
2
|{\nabla _z}(f)x|/{H_z}{(x,\bar x)^{1/2}}
, where x is a nonzero vector in
C
n
{\mathbb {C}^n}
and
H
z
{H_z}
is the Bergman metric on D. The number
β
f
(
z
)
{\beta _f}(z)
represents the maximum dilation of f at z. The set consisting of all
β
f
(
z
)
{\beta _f}(z)
for
z
∈
D
z \in D
and
f
:
D
→
Δ
f:D \to \Delta
holomorphic, is known to be bounded. We let
c
D
{c_D}
, be its least upper bound. In this work we calculate
c
D
{c_D}
for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which
β
f
=
c
D
{\beta _f} = {c_D}
) when D contains
Δ
\Delta
as a factor, and show that the class of extremal functions is very large when
Δ
\Delta
is not a factor of D.