The set of univalent functions as a subset of a Banach space (Q1330821)

Let \(B\) be the Banach space of functions \(f\) analytic in the open unit disk \(D\) with \(\| f \| = \sup_ D (| f(z) | (1 - | z |^ 2))\). The authors study the class \(S\) of normalized univalent functions in \(D\) as a subset of \(B\). A number of interesting results concerning isolated points and compactness for \(S\) and for some related function classes are derived. For example, it is shown that \(S\) has no isolated points and that the class \(C\) of normalized convex functions has no isolated points and is compact.