13 problems

A-parametric-representation conjecture. If $k_+(A)<2m(A)$ and $f(z,t)$ is an $A$-normalized univalent subordination chain satisfying the boundedness condition above, then…

Biholomorphic-chain conjecture. If $k_+(A)<2m(A)$ and $f(z,t)$ is the $A$-normalized univalent subordination chain given by the displayed limit, then $f(\cdot,t)$ is biholomorphic…

Let $\Gamma_\sigma$ denote the class of quasislits whose driving terms have the relevant bound $\sigma$, and let the Hölder exponent refer to capacity parametrization on intervals…

Let $\eta$ be a Loewner curve and let its regularity be measured in the class $C^{\alpha+1/2}$, with $C^{\alpha}$ understood using the usual convention for noninteger exponents. Lo…

Extension conjecture. Then $f\in{\mathcal{S}}^1$, that is, $f$ embeds into a normalized Loewner chain. The source states that this is equivalent to the generalized Andérsen–Lempert…

Embedding conjecture.

Let $\mathfrak S\subset\operatorname{Hol}(\mathbb{D},\mathbb{D})$ be a semigroup, and let $\mathfrak U$ denote the semigroup of univalent self-maps of $\mathbb{D}$. A semigroup adm…

Continuous-time Bedford conjecture. The Loewner range of $h(z,t)$ is biholomorphic to $\mathbb C^q$.

Let $S$ be the class of normalized univalent functions on the unit disc, and let $\Psi_n(S)$ denote its image under the Roper–Suffridge operator. A support point is a point at whic…

Let $S^0(\mathcal D_n)$ be the normalized Loewner class, and let $\varphi_{s,t}$ be an evolution family associated with an $\mathcal M(\mathcal D_n)$-Herglotz vector field. Bounded…

Let $S^0(\mathcal D_n)$ be the normalized Loewner class. Suppose that $f\in S^0(\mathcal D_n)$ is bounded by $M>0$, and let $G\in S^0(\mathcal D_n)$. Bounded-composition conjecture…

Let $S^0(\mathcal D_n)$ be the compact class of normalized mappings generated by Loewner chains, and call $f\in S^0(\mathcal D_n)$ a support point if a nonconstant real part of a c…

Let $\mathcal D_n$ denote the unit polydisc, and let $E(\mathcal D_n)$ and $F(\mathcal D_n)$ be the two classes of normalized mappings defined in the paper, with tilded versions al…