We extend our study of variability regions, Ali et al. [‘An application of Schur algorithm to variability regions of certain analytic functions–I’, Comput. Methods Funct. Theory, to appear] from convex domains to starlike domains. Let $\mathcal {CV}(\Omega )$ be the class of analytic functions f in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega }$ . As an application of the main result, we determine the variability region of $\log f'(z_0)$ when f ranges over $\mathcal {CV}(\Omega )$ . By choosing a particular $\Omega $ , we obtain the precise variability regions of $\log f'(z_0)$ for some well-known subclasses of analytic and univalent functions.

Suppose that [μ]T(Δ) is a point of the universal Teichmüller space T(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists μ such that μ is uniquely extremal in [μ]T(Δ) and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ]T(Δ) if [μ]T(Δ) admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

Much research has been done on the geometry of Teichmüller space and Hamilton sequences of extremal Beltrami differentials. This paper discusses some problems concerning infinitesimal Teichmüller geodesic discs and Hamilton sequences of extremal Beltrami differentials in the tangent space of an infinite-dimensional Teichmüller space.

We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.

Suppose that $T(S_0)$ is the Teichmüller space of a compact Riemann surface $S_0$ of genus $g>1$. Let $d_T(\cdot,\cdot)$ be the Teichmüller metric of $T(S_0)$ and let $d_S(\cdot,\cdot)$ be a metric of $T(S_0)$ defined by the length spectrums of Riemann surfaces. The author showed in a previous paper that $d_T$ and $d_S$ are topologically equivalent and $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2) \leq 2d_S(\tau_1,\tau_2)+C(\tau_1)$, where $C(\tau_1)$ is a constant depending on $\tau_1$. In this paper, it is shown that $d_T$ and $d_S$ are not metrically equivalent; that is, there is no constant $C>0$ such that $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2)\leq Cd_S(\tau_1,\tau_2)$ for all $\tau_1 {\rm and} \tau_2 {\rm in} T(S_0)$.