Let ψ be a conformal map of the unit disk D onto an unbounded domain and, for α > 0, let Fα = {z ∈ D : |ψ (z)| = α}. If H p (D) denotes the classical Hardy space and d D (0, Fα) denotes the hyperbolic distance between 0 and Fα in D, we prove that ψ belongs to H p (D) if and only if +∞ 0 α p−1 e −d D (0,Fα) dα < +∞. This result answers a question posed by P. Poggi-Corradini.

Let ψ be a conformal map on D with ψ (0)=0 and let Fα ={z ∈D:|ψ (z)|=α} for α>0. Denote by H p (D) the classical Hardy space with exponent p>0 and by h (ψ) the Hardy number of ψ. Consider the limitswhere ω D (0, Fα) denotes the harmonic measure at 0 of Fα and d D (0, Fα) denotes the hyperbolic distance between 0 and Fα in D. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and h (ψ)? Motivated by the result of Kim and Sugawa that h (ψ)=lim inf α→+∞ (log ω D (0, Fα) −1 log α), we show that h (ψ)=lim inf α→+∞ (d D (0, Fα)/log α). We also provide conditions for the existence of L and μ and for the equalities L=μ=h (ψ). Poggi-Corradini proved that ψ / ∈H μ (D) for a wide class of conformal maps ψ. We present an example of ψ such that ψ∈H μ (D).

We prove that a domain in the Riemann sphere is Gromov hyperbolic if and only if it is conformally equivalent to a uniform circle domain. This resolves a conjecture of Bonk-Heinonen-Koskela and also verifies Koebe's conjecture (Kreisnormierungsproblem) for the class of Gromov hyperbolic domains. Moreover, the uniformizing conformal map from a Gromov hyperbolic domain onto a circle domain is unique up to Möbius transformations. We also undertake a careful study of the geometry of inner uniform domains in the plane and prove the above uniformization and rigidity results for such domains.

Let D ⊂ C be a domain with 0 ∈ D. For R > 0, letωD (R) denote the harmonic measure of D ∩ {|z| = R} at 0 with respect to the domain D ∩ {|z| < R} and ωD (R) denote the harmonic measure of ∂D ∩ {|z| ≥ R} at 0 with respect to D. The behavior of the functions ωD andωD near ∞ determines (in some sense) how large D is. However, it is not known whether the functions ωD andωD always have the same behavior when R tends to ∞. Obviously, ωD (R) ≤ωD (R) for every R > 0. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant C such that for all simply connected domains D with 0 ∈ D and all R > 0,In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of D, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.

We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as 𝑝-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for the quasiconformal dilatation of the Beurling-Ahlfors extension of a homeomorphism of the real line, in terms of its symmetric distortion function. More specifically, the quasiconformal dilatation is bounded above by an average of the symmetric distortion function and below by the symmetric distortion function itself.As a consequence, the quasiconformal dilatation of the Beurling-Ahlfors extension of a homeomorphism of the real line is (sub)exponentially integrable, is 𝑝-integrable, or has a 𝐵𝑀𝑂 majorant if and only if the symmetric distortion is (sub)exponentially integrable, is 𝑝-integrable, or has a 𝐵𝑀𝑂 majorant, respectively. These theorems are all new and reconcile several sufficient extension conditions that have been established in the past.