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

Abstract

In this paper, a boundary version of the Carathéodory inequality is studied. For the function f(z), defined in the unit disc with f(0) = 0, R f(z) ≤ A, we estimate a modulus of angular derivative at the boundary point z0, Rf(z0) = A, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved.