The Caratheodory Inequality on the Boundary for Holomorphic Functions in the Unit Disc

Abstract

In this paper, a boundary version of the Caratheodory inequality is studied. For the function f(z), defined in the unit disc with f(0) = 0, Rf(z) <= A, we estimate a modulus of angular derivative at the boundary point z(0), Rf (z(0)) = A, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved.