As usual, we say that a function
f
∈
U
f \in U
if
f
f
is meromorphic in
|
z
|
>
1
| z | > 1
and has radial limits of modulus
1
1
a.e. (almost everywhere) on an arc
A
A
of
|
z
|
=
1
\left | z \right | = 1
. This paper contains three main results: First, we extend our solution of A. J. Lohwater’s problem (1953) by showing that if
f
∈
U
f \in U
and
f
f
has a singular point
P
P
on
A
A
, and if
υ
\upsilon
and
1
/
υ
¯
1/\bar {\upsilon }
are a pair of values which are not in the range of
f
f
at
P
P
, then one of them is an asymptotic value of
f
f
at some point of
A
A
near
P
P
. Second, we extend our solution of J. L. Doob’s problem (1935) from analytic functions to meromorphic functions, namely, if
f
∈
U
f \in U
and
f
(
0
)
=
0
f(0) = 0
, then the range of
f
f
over
|
z
|
>
1
\left | z \right | > 1
covers the interior of some circle of a precise radius depending only on the length of
A
A
. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on
|
z
|
=
1
\left | z \right | = 1
, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.