Let
S
\mathcal {S}
denote the standard normalized class of regular, univalent functions in
K
=
K
1
=
{
z
:
|
z
|
>
1
}
K = {K_1} = \{ z:|z| > 1\}
. Let
F
\mathcal {F}
be a given compact subclass of
S
\mathcal {S}
. We consider the following two problems. Problem 1. Find
max
|
a
2
|
\max |{a_2}|
for
f
∈
F
f \in \mathcal {F}
. Problem 2. For
|
z
|
=
r
>
1
|z| = r > 1
, find the
max
(
min
)
|
f
(
z
)
|
\max \;(\min )|f(z)|
for
f
∈
F
f \in \mathcal {F}
. In this paper we are concerned with the subclass
S
d
∗
(
M
)
=
{
f
∈
S
:
K
d
⊂
f
(
K
)
⊂
K
M
}
\mathcal {S}_d^\ast (M) = \{ f \in \mathcal {S}:{K_d} \subset f(K) \subset {K_M}\}
. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class
S
d
∗
(
M
)
\mathcal {S}_d^\ast (M)
, for all d, M such that
1
4
⩽
d
⩽
1
⩽
M
⩽
∞
\tfrac {1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty
.