ON BOUNDS FOR THE DERIVATIVE OF ANALYTIC FUNCTIONS AT THE BOUNDARY

Abstract

In this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for vertical bar f'(0)vertical bar and sharp lower bounds for vertical bar f'(c)vertical bar with c is an element of partial derivative D -{z:vertical bar z vertical bar - 1}. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function f(z) from below according to the second Taylor coefficients of f about z = 0 and z = z(0) not equal 0. Thanks to these inequalities, we see the relation between vertical bar f'(0)vertical bar and Rf(0). Similarly, we see the relation between Rf(0) and vertical bar f'(c)vertical bar for some c is an element of partial derivative D. The sharpness of these inequalities is also proved.