Grunsky-Nehari inequalities for a subclass of bounded univalent functions

Abstract

Let D 1 {D_1} be the class of regular analytic functions F ( z ) F(z) in the disc U = { z : | z | > 1 } U = \{ z:|z| > 1\} for which F ( 0 ) > 0 , | F ( z ) | > 1 F(0) > 0,|F(z)| > 1 , and F ( z ) + F ( ζ ) ≠ 0 F(z) + F(\zeta ) \ne 0 for all z , ζ ∈ U z,\zeta \in U . Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in D 1 {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 {D_1} and the class S 1 {S_1} of bounded univalent functions, the results can be reinterpreted to apply to this latter class.