A sharp Carathéodory's inequality on the right half plane

Abstract

. In this paper, a boundary version of Carath´eodory’s inequality on the right half plane is investigated. Here, the function Z ( s ) , is given as Z ( s ) = 1 + c 1 ( s − 1 )+ c 2 ( s − 1 ) 2 + ... be an analytic in the right half plane with ℜ Z ( s ) (cid:2) A ( A > 1 ) for ℜ s (cid:3) 0. We derive inequalities for the modulus of Z ( s ) function, | Z (cid:2) ( 0 ) | , by assuming the Z ( s ) function is also analytic at the boundary point s = 0 on the imaginary axis and fi nally, the sharpness of these inequalities is proved.