On certain subclasses of analytic and harmonic mappings

TL;DR

This paper studies subclasses of harmonic and analytic functions in the unit disk, deriving properties like coefficient bounds, growth estimates, and sharp bounds of the second Hankel determinant for certain normalized functions.

Contribution

It introduces and analyzes a new class of harmonic functions with specific conditions, providing fundamental properties and sharp bounds for inverse logarithmic coefficients.

Findings

Coefficient bounds for functions in the class _{\u00a0H}}^0(lpha, M)

Growth estimates and starlikeness properties established

Sharp bound of the second Hankel determinant for certain univalent functions

Abstract

Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-\alpha)h'(z)+\alpha zh''(z)-1+\alpha\right|\leq M+\left|(1-\alpha)g'(z)+\alpha zg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $\alpha\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$.