We shall show that Beurling-Tsuji’s theorem (see Theorem A) is, in a sense, best possible. For each pair
a
,
b
∈
(
0
,
+
∞
)
a, b \in (0, + \infty )
there exists a function f holomorphic in
{
|
z
|
>
1
}
\{ |z| > 1\}
such that the Euclidean area of the Riemannian image of each non-Euclidean disk of non-Euclidean radius a, is bounded by b, and such that f has finite angular limit nowhere on the unit circle.