Communications of the
Korean Mathematical Society CKMS

In this paper, a generalized boundary version of Carath\'{e}od\-ory's inequality for holomorphic function satisfying $f(z)=f(0)+a_{p}z^{p}+\cdots,$ and $\Re f(z)\leq A$ for $\left\vert z\right\vert <1$ is investigated. Also, we obtain sharp lower bounds on the angular derivative $f^{\prime }(c)$ at the point $c$ with $\Re f(c)=A$. The sharpness of these estimates is also proved.

Keywords: holomorphic function, Schwarz lemma on the boundary, Cara\-th\'{e}odory's inequality

MSC numbers: Primary 30C80, 32A10