Julia's lemma is a boundary version of the Schwarz lemma, where the fixed point is located on the unit circle. Given the angular derivative at a boundary point, it determines the range of values the function can take inside the disk. In this talk, we show that the family of functions with a prescribed positive angular derivative at a boundary point is not compact, and we investigate the structure of its compactification. Based on these observations, we establish a sharp distortion estimate for such families.
