7 problems

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…

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 $S^0(\mathcal D_n)$ 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…