Conformal invariants and the angular derivative problem

Abstract

We prove necessary and sufficient conditions for the existence of an angular derivative for a simply connected domain Ω ⊊ ℂ. Our conditions involve hyperbolic distances, Green functions, and harmonic measures. We use these conditions to construct two classes of domains possessing an angular derivative. M S C ( 2 0 2 0 )30C35, 31A15, 30F45, 30C85 (primary) INTRODUCTIONLet 𝑓 be holomorphic in the unit disk 𝔻. We say that 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 ∈ 𝜕𝔻 if 𝑓(𝑧) → 𝑓(𝜁), as 𝑧 → 𝜁, in each Stolz angle 𝑆(𝜁) with vertex at 𝜁. We say that 𝑓 is semi-conformal at 𝜁 if 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 and arg 𝑓(𝑧) − 𝑓(𝜁) 𝑧 − 𝜁 has finite non-tangential limit as 𝑧 → 𝜁. Some authors use the term isogonal instead of semiconformal. If 𝑓 is semi-conformal at 𝜁, then every smooth arc ending at 𝜁, non-tangentially to 𝜕𝔻, is mapped onto a smooth arc ending at 𝑓(𝜁) and the angles between such arcs are preserved. See, for example, [20]. We say that 𝑓 has angular derivative 𝑓 ′ (𝜁) at 𝜁 if 𝑓 has non-tangential limit 𝑓(𝜁) at 𝜁 and 𝑓(𝑧) − 𝑓(𝜁) 𝑧 − 𝜁 → 𝑓 ′ (𝜁), as 𝑧 → 𝜁, 𝑧 ∈ 𝑆(𝜁),for every Stolz angle 𝑆(𝜁). If 𝑓 ′ (𝜁) is finite and non-zero, then 𝑓 is semi-conformal at 𝜁.

Beurling-Carleson sets, inner functions and a semi-linear equation

“…Rodin and Warschawski gave an if and only if condition for ψ to possess an angular derivative at p in terms of moduli of curve families, e.g. see [GM05, Theorem V.5.7] or [BK22]. When Ω is a starlike domain with regular boundary, their condition takes a simpler form [IK22]:…”