Grunsky-Nehari inequalities for a subclass of bounded univalent functions

Abstract:

Let ${D_1}$ be the class of regular analytic functions $F(z)$ in the disc $U = \{ z:|z| < 1\}$ for which $F(0) > 0,|F(z)| < 1$, and $F(z) + F(\zeta ) \ne 0$ for all $z,\zeta \in U$. Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in ${D_1}$, the proof being based on the area method. By subordination it is shown univalency is unnecessary for certain special cases of the inequalities. Employing a correspondence between ${D_1}$ and the class ${S_1}$ of bounded univalent functions, the results can be reinterpreted to apply to this latter class.