A Theorem of Beurling and Tsuji is Best Possible
Shinji Yamashita · Proceedings of the American Mathematical Society · 1978
We shall show that Beurling-Tsuji’s theorem (see Theorem A) is, in a sense, best possible. For each pair $a, b \in (0, + \infty )$ there exists a function f holomorphic in $\{ |z| < 1\}$ 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.