Univalent functions in the Banach algebra of continuous functions (Q394714)

Let \(C(T)\) denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric space \(T\) and \(B_0\) be the open unit ball of \(C(T)\). For a given function \(\phi\) in the class \(\mathcal{S}\) of univalent functions on the open unit disk of the complex plane, the authors consider the composition operator \(F_\phi : B_0 \to C(T)\) defined by \( F_\phi(f) = \phi \circ f\) and introduce the class \(S_C\) of all functions \(F_\phi\) which are injective in \(B_0\). They prove that \( F_\phi\in S_C\) if and only if \(\phi \in S\). As a corollary, it is shown that \(|a_n| \leq n \) for \[ F_\phi(f) = f +\sum_{n=1}^\infty a_nf^n \in S_C. \] If the derivative of a normalized analytic function defined on a convex domain has positive real part, then the well-known Noshiro-Warschawski theorem says that the function is univalent. This result is extended to the functions defined on \(B_0\). It is also shown that the set \(\{F_\phi(f) : f \in B_0\}\) is a convex set for any extreme point \(\phi\) of the class of normalized convex functions defined on \(\mathbb{D}\). The authors obtain an estimate for \(\|F_\phi^{(n)}(f)\|\).