Analyticity on rotation invariant families of curves

Abstract

Let G \mathfrak {G} be a rotation invariant family of smooth Jordan curves contained in Δ \Delta , the open unit disc in C {\mathbf {C}} . For each Γ ∈ G \Gamma \in \mathfrak {G} let D Γ {D_\Gamma } be the simply connected domain bounded by Γ \Gamma . We present various conditions which imply that if f f is a continuous function on Δ \Delta such that for every Γ ∈ G \Gamma \in \mathfrak {G} the function f | Γ f|\Gamma has a continuous extension to D Γ ¯ \overline {{D_\Gamma }} which is analytic in D Γ {D_\Gamma } , then f f is analytic in Δ \Delta .