Semiduality of small sets of analytic functions (Q1409779)

Let \(A\) be the set of analytic functions \(f\) in \(D=\{z\mid| z|< 1\}\) for which \(f(0)= 1\). Let \(f* g\) denote the convolution of \(f\) and \(g\) in \(A\). If \(V\) is a subset of \(A\), the dual set \(V\) is the set \(\{g\in A\mid\forall f\in V\;\forall z\in D(f*g)(z)\neq 0\}\). \(V\) is termed dual if \(V= (V^*)^*\). For \(f\in A\) and \(x\in\overline D\), let \(f_x(z)= f(xz)\) for \(z\in D\), and let \(A\) have the topology of uniform convergence on compact subsets of \(D\). A subset \(V\) of \(A\) is complete if it equals its completion \(V^C= \{f_x\mid f\in V,| x|\leq 1\}\). A set \(V\) in \(A\) is termed semidual if \(V^C\) is dual. \textit{V. Kasten} and \textit{St. Ruscheweyh} [Math. Nachr. 123, 277--283 (1985; Zbl 0589.30010)] have determined a so-called neighborhood property of \(V^{**}\) which implies \(V\) is non-semidual. After some initial discussion on semiduality, the paper under review focuses on examples of non-semidual subsets of \(A\) consisting of two functions. The proofs follow the pattern that the set \(V\) imposes restrictions on \(V^*\) from which may be shown either the normality of \(V^*\) or the boundedness of certain coefficients in an expansion for \(g\) in \(V^*\) using normality criteria. These imply \(V^{**}\) has the neighborhood property. Some of the strongest results on normal families are used. Earlier work, such as \textit{St. Ruscheweyh} and \textit{K.-J. Wirths} [Result. Math. 10, 147--151 (1986; Zbl 0623.30044)], \textit{H. Pinto}, \textit{St. Ruscheweyh} and \textit{L. Salinas} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 40, 193--207 (1986; Zbl 0654.30016)], \textit{K.-J. Wirths} [Pitman Res. Notes Math. Ser. 262, 106--112 (1992; Zbl 0797.30007)] and \textit{J. Grahl} [J. Anal. Math. 82, 207--220 (2000; Zbl 0970.30003)], discusses semiduality of subsets of \(A\) consisting of single functions.