Some Remarks on Schwarz Lemma at the Boundary

In this paper, a boundary version of the Schwarz lemma is investigated. We consider a function f holomorphic in the unit disc D, f(a) = b, broken vertical bar a broken vertical bar < 1 and broken vertical bar f(z) - alpha broken vertical bar < alpha for z is an element of D, where a is a positive real number and 1/2 < alpha <= 1. We obtain sharp lower bounds on the angular derivative f'(c) at the point c, where f(c) = 2 alpha, broken vertical bar c broken vertical bar = 1.