Inequalities for angular derivatives at the boundary of the unit disc

Abstract

In this paper, a boundary version of Schwarz inequality is investigated. For the function f(z)=β + c_pz^p + ... defined in the unit disk, ℜf(z) > α, 0 ≤ α < β ( β is any integer ≥ 1), we estimate a modulus of angular derivative at the boundary point z₀, ℜf(z₀)=α, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.