Tangential limits of starlike univalent functions

Abstract

Let f f be starlike univalent in the unit disc, let γ > 1 \gamma > 1 and let K > 0 K > 0 . Then f ( z ) f(z) tends to a limit as z → e i θ z \to {e^{i\theta }} inside { z : | e i θ − z | ≤ K ( 1 − | z | ) 1 / γ } \{ z:\left | {{e^{i\theta }} - z} \right | \leq K{(1 - \left | z \right |)^{1/\gamma }}\} for every θ \theta in [ 0 , 2 π ] [0,2\pi ] . This result is sharp.