The Carathéodory inequality on the boundary for holomorphic functions in the unit disc

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