On certain subclasses of analytic and harmonic mappings

TL;DR

This paper introduces the harmonic subclass $\mathcal{D}_{\mathcal{H}}^0(\alpha,M)$ and derives sharp coefficient bounds, sharp growth estimates, and a Jacobian bound, while also establishing a correspondence with the analytic class $\mathcal{D}(\alpha,M)$. It then analyzes the second Hankel determinant of logarithmic inverse coefficients for functions in the analytic class $\mathcal{P}(M)$, obtaining sharp, piecewise bounds for $0<M\le 1/\log4$ and identifying extremal functions via Carathéodory parameterization. The approach combines subordination, hypergeometric function bounds, and detailed coefficient lemmas to yield precise extremal results for both harmonic mappings and inverse logarithmic coefficients. The results enhance understanding of geometric properties of structured harmonic mappings and provide exact bounds for inverse coefficient functionals within these subclasses.

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 $\mathcal{D}_{\mathcal{H}}^0(α, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-α)h'(z)+αzh''(z)-1+α\right|\leq M+\left|(1-α)g'(z)+αzg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $α\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(α, M)$, 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 $z\in\mathbb{D}$.

Figures (12)

• Figure 1: The graph of $(2-3p_1+ 2p_1^2)/(3p_1^2)$ for $p_1\in\left(0,1\right)$
• Figure 2: The graph of $(-2+3p_1)/(3p_1^2)$ for $p_1\in\left(0,1\right)$
• Figure 3: The graph of the polynomial $\Psi(M)$ for $1/3<M\leq 1/\log4$
• Figure 4: The graph of the polynomials $\Psi_1(M)$ and $\Psi_2(M)$ for $0<M<1/3$
• Figure 5: The graph of the polynomial $\phi_1(M)$ for $0<M< 1/6$

Theorems & Definitions (19)

• Theorem 2.1
• proof
• Remark 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.2
• Theorem 2.4
• proof