Let G be a closed subset of the closed unit disc in C, let F be a closed subset of the unit circle of measure 0 and let
Φ
\Phi
map G into the class of all open subsets of a complex Banach space X. Under suitable additional assumptions on
Φ
\Phi
we prove that given any continuous function
f
:
F
→
X
f:\,F \to X
satisfying
f
(
z
)
∈
closure(
Φ
(
z
)
)
(
z
∈
F
∩
G
)
f(z)\, \in \,{\text {closure(}}\Phi (z))\,(z\, \in \,F\, \cap \,G)
there exists a continuous function f from the closed unit disc into X, analytic in the open unit disc, which extends f and satisfies
f
~
(
z
)
∈
Φ
(
z
)
(
z
∈
G
−
F
)
\tilde f(z)\, \in \,\Phi (z)\,(z\, \in \,G\, - \,F)
. This enables us to generalize and sharpen known dominated extension theorems for the disc algebra.