M. Levenshtein | Searcharxiv

arXiv2013

Boundary distortion estimates for holomorphic maps

We establish some estimates of the the angular derivatives from below for holomorphic self-maps of the unit disk at one and two fixed points of the unit circle provided there is no fixed point inside the unit disk. The results complement Cowen-Pommerenke and Anderson-Vasil'ev type estimates in the case of univalent functions. We use the method of extremal length and propose a new semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to receive a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.

math.CV↗

arXiv2005

Rigidity of holomorphic generators and one-parameter semigroups

In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $τ$ of the open unit disk $Δ$. Namely, if $f\in\mathrm{Hol}(Δ,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_{t}\}_{t\geq0}$, we state that the equality $f(z)=o(|z-τ|^{3})$ when $z\toτ$ in each non-tangential approach region at $τ$ implies that $f$ vanishes identically on $Δ$. Note, that if $F$ is a self-mapping of $Δ$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz. We also prove that two semigroups $\{F_{t}\}_{t\geq0}$ and $\{G_{t}\}_{t\geq0}$, with generators $f$ and $g$ respectively, commute if and only if the equality $f=αg$ holds for some complex constant $α$. This fact gives simple conditions on the generators of two commuting semigroups at their common null point $τ$ under which the semigroups coincide identically on $Δ$.

math.CV↗