APPLICATIONS OF JACK'S LEMMA FOR CERTAIN SUBCLASSES OF HOLOMORPHIC FUNCTIONS ON THE UNIT DISC

Abstract

In this paper, we give some results on zf' (z)/f(z) for the certain classes of holomorphic functions in the unit disc E = {z : vertical bar z vertical bar< 1} and on partial derivative E {z : vertical bar z vertical bar = 1}. For the function f (z) = z(2) + c(3)z(3) + c(4)z(4) + ... defined in the unit disc E such that f (z) is an element of A(alpha), we estimate a modulus of the angular derivative of zf' (z)/f(z) function at the boundary point b with bf' (b)/f(b) = 1 + alpha. Moreover, Schwarz lemma for class A(alpha) is given. The sharpness of these inequalities is also proved.