Let ${\mathbb D}$ be the open unit disk, and let $\mathcal {A}(p)$ be the class of functions f that are holomorphic in ${\mathbb D}\backslash \{p\}$ with a simple pole at $z=p\in (0,1)$, and $f'(0)\neq 0$. In this article, we significantly improve lower bounds of the Bloch and the Landau constants for functions in ${\mathcal A}(p)$ which were obtained in Bhowmik and Sen (2023, Monatshefte für Mathematik, 201, 359–373) and conjecture on the exact values of such constants.

For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $ , we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$ , denoted by $\mathcal {D}^s_K$ . Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$ , we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Following the investigations of $\text{B}$. Abrahamse $[1]$, F. Forelli $[11]$, M. Heins $[14]$ and others, we continue the study of the Pick-Nevanlinna interpolation problem in multiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.

We study the asymptotic behavior of the n-widths of a class of entire functions in weighted approximation on subsets of the complex plane.