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Rajesh Hossain

Publications and source records attributed to Rajesh Hossain.

13 recordsLinked to original sources
arXiv2026

Schwarzian norm estimates for some classes of analytic and harmonic mappings

Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{} < 1\}$. For $\beta > 1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\} < \beta$ for $z \in \mathbb{D}$. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm $\Vert{}P_f\Vert{}$ and Schwarzian norm $\Vert{}S_f\Vert{}$ for functions $f \in \mathcal{N}(\beta)$, parametrized by $f''(0)$, with special emphasis on the case $f''(0) = 0$. In addition, sharp growth, distortion, and radius results (convexity and concavity) for $\mathcal{N}(\beta)$ are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings $f = h + \bar{g}$ whose analytic part $h$ belongs to $\mathcal{N}(\beta)$.

math.CV↗
arXiv2026

Bohr-Type Inequalities for Shifted Disks via Optimal $H^2$-Embeddings

The primary objective of this paper is to systematically generalize this phenomenon by replacing the standard unit disk with a family of nested, internally tangent shifted disks $\Omega_\gamma$ parameterized by $\gamma \in [0, 1)$, defined by$$\Omega_\gamma = \left\{ z \in \mathbb{C} : \left| z + \frac{\gamma}{1 - \gamma} \right| < \frac{1}{1 - \gamma},\; \gamma\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $\Omega_\gamma$ and evaluating the limiting behavior as $\gamma \to 1^-$, we establish a novel framework to determine the Bohr radius for the unbounded half-plane $\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$. Furthermore, we prove several sharp variations of the Bohr inequality within these domains, including refined and improved formulations for unimodular bounded analytic functions. The results obtained herein not only extend classical radius problems to unbounded regions but also illuminate the delicate interplay between domain deformation and coefficient estimates.

math.CV↗
arXiv2026

Bohr, Bohr-Rogosinski, and Landau-Type Results for a Generalized Class of Harmonic Mappings

In this paper, we study the Bohr phenomenon for a generalized subclass of harmonic mappings defined by a second-order differential inequality in the unit disk. Specifically, we consider the class $\mathcal{BH}_0(\gamma, \delta)$, which extends several known subclasses of harmonic and analytic functions. By employing sharp coefficient estimates and growth results, we establish improved versions of Bohr-type inequalities, including refined Bohr radii and Bohr--Rogosinski radii for this class. Furthermore, we derive generalized inequalities involving higher-order coefficient sums and area terms, thereby extending classical Bohr inequalities in a harmonic setting. The sharpness of the obtained results is verified through extremal functions. In addition, we obtain Landau-type theorems for the class $\mathcal{BH}_0(\gamma, \delta)$, providing explicit bounds for the radius of univalence and the size of schlicht disks contained in the image domain. Our results not only unify and extend several earlier works but also provide new insights into the geometric behavior of harmonic mappings under differential constraints.

math.CV↗
arXiv2026

Bohr Radius and Landau-type Theorems for Harmonic Mappings with Boundary Functions in Lebesgue Spaces

This paper investigates the geometric and analytical properties of harmonic mappings $f$ in the unit disk $\mathbb{D}$ induced by boundary functions $F$ belonging to the Lebesgue spaces $L^{p}(\mathbb{T})$ for $1 \le p \le \infty$. We first establish a sharp Bohr-type inequality for the class of bounded harmonic mappings. Specifically, we prove that for a fixed analytic part $|a_{0}|= aM$, the majorant series $M_{f}(r)$ satisfies $M_{f}(r) \le M$ for $r \le (1-a)/(1-a+4/\pi)$, and demonstrate that this radius is best possible. This result is subsequently extended to harmonic mappings with $L^p$ boundary functions, where we determine the sharp Bohr radius $r_{p} = 1/(2C_{q}+1)$, with $C_{q}$ being a constant depending on the conjugate exponent $q$. Furthermore, the paper provides improved Landau-type theorems for these mappings. Under standard normalization, we derive explicit expressions for the radius of univalence $r_{0}$ and the radius of the inscribed schlicht disk $R_{0}$. The sharpness of these constants is discussed through the construction of extremal functions related to the Poisson kernel.

math.CV↗
arXiv2026

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

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 $h(0)=0=h'(0)-1$, 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 $\mathcal{W}_{\mathcal{H}}^{0}(\alpha)$ for $\alpha \ge 0$. For mappings in $\mathcal{W}_{\mathcal{H}}^{0}(\alpha)$, 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 $\Phi(z,s,a)$ and the Dilogarithm function ${\rm Li}_2(z)$. 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.

math.CV↗
arXiv2026

Geometric subfamily of locally univalent functions, Blaschke products and quasidisk

In this article, we consider the family $\mathcal{F}(\alpha)$ defined for $\alpha \in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \frac{\alpha}{2} \quad \text{for } z \in \mathbb{D}. \end{align*} Our primary objective is to show that this family possesses significant geometric and analytic properties, including connections with Blaschke products and the Schwarzian derivative, as well as its sharp bounds. Furthermore, we prove that if $f \in \mathcal{F}(\alpha)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(\alpha)$, then $\|S_f\| = 2\alpha(2-\alpha)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2\alpha+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(\alpha)$, where the analytic part $h$ belongs to $\mathcal{F}(\alpha)$.

math.CV↗
arXiv2025

Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class

This paper investigates the geometric properties of functions within the generalized Robertson class which consists of alpha-starlike functions of order beta. The study's significance lies in providing a deeper understanding of the univalence and geometric behavior of these functions, which are fundamental in complex analysis and geometric function theory. The primary objective is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in this class. These bounds are expressed in terms of the initial coefficient of the function, specifically focusing on the important case where this initial coefficient is zero. Additionally, the paper establishes sharp distortion and growth theorems for the functions belonging to this generalized class. Finally, the research addresses the radius problem for this function class by determining the sharp radius of concavity and the sharp radius of convexity.

math.CV↗
arXiv2025

Pre-Schwarzian and Schwarzian norm Estimates for Robertson class

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-\pi/2<\alpha<\pi/2$, let $\mathcal{S}_{\alpha}$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{i\alpha}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}_{\alpha}$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$.

math.CV↗
arXiv2025

Schwarzian Norm Estimates for Analytic Functions Associated with Convex Functions

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disc $\mathbb{D}=\{z\in\mathbb{C}:\;|z|<1\}$ normalized by $f(0)=0$ and $f^{\prime}(0)=1$. In the present article, we consider and $\mathcal{F}(c)$ the subclasses of $\mathcal{A}$ are defined by \begin{align*} \mathcal{F}(c)=\bigg\{f\in\mathcal{A}:\;{\rm Re}\;\bigg(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg)>1-\frac{c}{2},\;\;\mbox{for some}\;c\in(0,3]\bigg\}, \end{align*} and derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in and $\mathcal{F}(c)$ expressed in terms of their value $f^{\prime\prime}(0)$, in particular, when the quantity is equal to zero. Moreover, we obtain sharp bounds for distortion and growth theorems for functions in the class $\mathcal{F}(c)$.

math.CV↗
arXiv2024

Pre-Schwarzian and Schwarzian norm Estimates for class of Ozaki Close-to-Convex functions

The primary objective of this paper is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives in the Ozaki close-to-convex functions $f$, expressed in terms of their value $f^{\prime\prime}(0)$, in particular, when the quantity is equal to zero. Additionally, we obtain sharp bounds for distortion and growth theorems. We will also derive the sharp bound of pre-Schwarzian norm for a certain class of harmonic mappings whose analytic part is fixed.

math.CV↗
arXiv2024

Sharp radius of concavity for certain classes of analytic functions

Let $\mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $\mathbb{D}$ with the normalization $f(0)=0=f^{\prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$, namely $\mathcal{S}_0^{(n)}$, $\mathcal{K(α,β)}$, $\mathcal{\tilde{S^*}(β)}$, and $\mathcal{S}^*(α)$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible.

math.CV↗
arXiv2024

Radius of concavity for certain class of functions

Let $ \mathcal{S}(p) $ be the class of all meromorphic univalent functions defined in the unit disc $ \mathbb{D} $ of the complex plane with a simple pole at $ z=p $ and normalized by the conditions $ f(0)=0 $ and $ f^{\prime}(0)=1 $. In this paper, we find radius of concavity and compute the same for functions in $ \mathcal{S}(p) $ and for some other well-known classes of functions on unit disk. We explore general linear combinations $F(z):=\lambda_1f_1(z)+\cdots+\lambda_{2n} f_{2n}(z),\; \lambda_j\in\mathbb{C} $, $ n\in\mathbb{N} $, of functions belonging to the class $\mathcal{S}(p)$ and some other classes of functions of analytic univalent functions and investigate their radii of univalence, convexity and concavity.

math.CV↗