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, ?f(z) ? A, we estimate a modulus of angular derivative at the boundary point z0, ?f(z0) = A, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved. © B.N. Örnek, 2016.