INEQUALITIES FOR THE ANGULAR DERIVATIVES OF CERTAIN CLASSES OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISC

Abstract

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function f (z) - z+c(p+1)z(p+1)+c(p+2)z(p+2)+... holomorphic in the unit disc and vertical bar f (z)/lambda f(z)+(1-lambda)z - alpha vertical bar < alpha for vertical bar z vertical bar < 1, where 1/2 < alpha <= 1/1+lambda, 0 <= lambda < 1. If we know the second and the third coefficient in the expansion of the function f (z) = z + c(p+1)z(p+1) + c(p+2) z(p+2) +...., then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account c(p+1), c(p+2) and zeros of f(z)-z. We obtain a sharp lower bound of vertical bar f '(b)vertical bar at the point b, where vertical bar b vertical bar = 1.