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 ${\mathcal{S}}^{\ast }$ of starlike functions and give a simple explicit example of a function in $\mathcal{R}\setminus {\mathcal{S}}^{\ast }$.