Shota Hoshinaga | Searcharxiv

arXiv2026

Boundary geometry and linear accessibility of functions with positive real derivative

We study the boundary geometry of the Noshiro--Warschawski class $\mathcal{R}$. Using the geometric structure of close-to-convex domains and their relation to Loewner chains, we investigate the boundary behavior of functions in $\mathcal{R}$. In particular, we discuss the relation between spherical length and local connectedness, and show that the boundary of the image domain of a function in $\mathcal{R}$ need not be locally connected. We also revisit the classical fact that $\mathcal{R}$ is not contained in the class $\mathcal{S}^{*}$ of starlike functions and give a simple explicit example of a function in $\mathcal{R}\setminus\mathcal{S}^*$.

math.CV↗

arXiv2022

Continuous evolution families

Recently in relation to the theory of non-commutative probability, a notion of evolution families $\{ω_{s,t}\}_{s \le t}$ is generalized that are only continuous in parameters, namely $(s,t) \mapsto ω_{s,t}$ is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.

math.CV↗