Landau-type Theorems for Polyanalytic and Log-$α$-analytic functions
P. Li, M. -S. Liu, S. Ponnusamy, H. Zhao
TL;DR
The paper addresses Landau-type univalence for polyanalytic and log-$\alpha$-analytic functions by deriving new coefficient estimates for bounded polyanalytic maps and establishing three versions of Landau-type theorems in the polyanalytic setting. A key monotonicity lemma and an extension of a lemma from prior work are developed and used to obtain explicit univalence radii $r$ and schlicht-disk radii $\sigma$ (e.g., $r_1$ as the least positive root of a specific equation and $\sigma_1$ given in terms of $M$ and $\alpha$). The results are then extended to $\log$-$\alpha$-analytic functions, yielding three corresponding Landau-type results and sharp corollaries, including univalence on the full disk when $|\log f(z)|\le 1$ and an explicit Bloch-type constant for such mappings. Collectively, these findings extend Landau-type theory from analytic and polyharmonic cases to polyanalytic and log-$\alpha$-analytic classes, with concrete radii and image-disk guarantees useful for univalence and distortion analyses.
Abstract
In the present article, we investigate the univalence property of polyanalytic functions and $\log$-$α$-analytic functions. First, by using a new idea, we prove an improved lemma and the coefficient estimates for bounded polyanalytic functions on the unit disk. Then, we present three versions of Landau-type theorems for such functions and determine the univalence domain and the radius of schlicht disk. Finally, as a consequence, the Landau-type theorems for $\log$-$α$-analytic functions are also provided.