Under a certain assumption of
f
f
and
g
g
in
L
∞
{L^\infty }
which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra
[
H
∞
,
f
]
[{H^\infty },\,f]
generated by
H
∞
{H^\infty }
and
f
f
. It is proved that (1) ball
(
B
/
H
∞
+
C
)
(B/{H^\infty } + C)
does not have exposed points for every Douglas algebra
B
B
, (2) Sarason’s three functions problem is solved affirmatively, (3) some characterization of
f
f
for which
[
H
∞
,
f
]
[{H^\infty },\,f]
is singly generated, and (4) the
M
M
-ideal conjecture for Douglas algebras is not true.