The dual and dual hulls of classes of analytic functions (Q1589158)

\textit{St. Rusheweyh}, [Trans. Am. Math. Soc. 210, 63-74 (1975; Zbl 0311.30011)], introduced and studied the properties of the dual sets to the given class \(V\) denoted by \(V^*\) and defined as follows \[ V^*=\bigl \{g\in A_0(D): (f*g)(z)\neq 0 \text{ for every }f\in V\text{ and }z\in D\bigr\}, \] where \(A_0(D)\) is the class of functions regular in \(D=\{z:|z|<1\}\), \(f(0)=1\), \(V\) is a given subclass of \(A_0(D)\) and \(f*g\) denotes the convolution (Hadamard product) of the functions \(f\) and \(g\). In that paper he proves also that if \(V\) is compact and such that \(f\in V\Rightarrow f(xz)\in V\) for every \(x\), \(|x|\leq 1\), than for every linear continuous functional \(\lambda\) we have \[ \lambda (V^{**}) =\lambda(V), \text{ where }V^{**}= (V^*)^*. \] In this paper the author investigates similar problems. He gives some weaker conditions on the subclasses \(V\) such that the relation \(\lambda(V^{**}) =\lambda(V)\) are still satisfied for every linear continuous functional \(\lambda\) defined on \(A_0(D)\). He also gives the new definition of \(V^*\) and \(V^{**}\). For examples it is proved that if \(V\subset A_0(D)\) is compact then \(V^*= \overline{(cm(V))^T}\), where \(cm(V)= \{h(z)=f(xz): f\in V,|x|\leq 1\}\), \(V^T=\{g\in A_0(\overline D):(f*g) (1)\neq 0\) for every \(f\in V\}\).