Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc

Abstract:

Let $A$ be the set of functions regular in the unit disc $\mathcal {U}$ and ${A_0}$ the set of all functions $f \in A$ which satisfy $f(0) = 1$. For $V \subset {A_0}$ define the dual set ${V^ \ast } = \{ f \in {A_0}|f \ast g \ne 0{\text { for all }}g \in V,z \in \mathcal {U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }$. Here $f \ast g$ denotes the Hadamard product. THEOREM. Let $V \subset {A_0}$ have the following properties: (i) $V$ is compact, (ii) $f \in V$ implies $f(xz) \in V$ for all $|x| \leqslant 1$. Then $\lambda (V) = \lambda ({V^{ \ast \ast }})$ for all continuous linear functionals $\lambda$ on $A$. This theorem has many applications to functions in $A$ which are defined by properties like bounded real part, close-to-convexity, univalence etc.