ESTIMATES FOR SECOND NON-TANGENTIAL DERIVATIVES AT THE BOUNDARY

Abstract

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f(z) holomorphic in the unit disc and f (0) = 0, f'(0) = 1 such that R f'(z) > 1-alpha/2, -1 < alpha < 1, we estimate a modulus of the second non-tangential derivative of f(z) function at the boundary point z(0) with Rf'(z(0)) = 1-alpha/2, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below vertical bar f ''(z(0))vertical bar according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z(1) not equal 0. The sharpness of these inequalities is also proved.