Communications of the
Korean Mathematical Society CKMS

In this paper, a boundary version of Carath\'{e}odory's inequality on the right half plane for $p$-valent is investigated. Let $Z(s)=1+c_{p}\left( s-1\right) ^{p}+c_{p+1}\left( s-1\right) ^{p+1}+\cdots$ be an analytic function in the right half plane with $\Re Z(s)\leq A$ $\left( A>1\right) $ for $\Re s\geq 0$ . We derive inequalities for the modulus of $Z(s)$ function, $|Z^{\prime }(0)|$, by assuming the $Z(s)$ function is also analytic at the boundary point $s=0$ on the imaginary axis and finally, the sharpness of these inequalities is proved.

Keywords: Schwarz lemma on the boundary, Carath\'{e}odory's inequality, analytic function

MSC numbers: Primary 30C80, 32A10