Sharp Landau-Type Theorems and Schlicht Disc Radii for certain Subclasses of Harmonic Mappings

Abstract

Let $\mathcal{H}$ be the class of all complex-valued harmonic mappings $f=h+\overline{g}$ defined on the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalization $h(0)=0=h^{\prime }(0)-1$, here $h$ and $g$ are analytic functions in $\mathbb{D}$. In this paper, we investigates Landau-type theorems for several significant subclasses of sense-preserving harmonic mappings. Specifically, we establish sharp Landau-type theorems for the class ${\mathcal{P}}_{\mathcal{H}}^0(M)$ and the parameterized class ${\mathcal{W}}_{\mathcal{H}}^0(\hat{I}\pm )$ for $\hat{I}\pm \ge 0$. For mappings in ${\mathcal{W}}_{\mathcal{H}}^0(\hat{I}\pm )$, we derive the radii of univalence and the radii of the largest schlicht discs contained in the images of the unit disc, expressing these results in terms of the Lerch Transcendent function $\hat{I}¦(z,s,a)$ and the Dilogarithm function ${\mathrm{Li}}_2(z)$. The sharpness of the obtained radii is demonstrated by constructing appropriate extremal functions for each class. These results generalize and extend various known Landau-type theorems in the theory of harmonic mappings.

23 pages, 4 figures