arXiv subjects
Publications and source records attributed to Molla Basir Ahamed.
In this paper, we study the Bohr inequality with lacunary series for vector-valued holomorphic functions defined in unit ball of finite dimensional Banach sequence space. Also, we study the Bohr-Rogosinski inequality for same class of functions. All the results are proved to be sharp.
In this paper, we prove several sharp Bohr-type and Bohr-Rogosinski-type inequalities for $K$-quasiconformal, sense-preserving harmonic mappings on $\mathbb{D}$, whose analytic part is subordinate to a function belonging to the class of concave univalent functions on $\mathbb{D}$. In addition, we derive Bohr-type inequalities for $K$-quasiconformal, sense-preserving harmonic mappings on $\mathbb{D}$, where the analytic part is subordinate to a function from the Ma-Minda class of convex and starlike functions. The results generalize several existing results.
A crucial extension of quaternionic function theory to octonions is the concept of slice regular functions, introduced to handle holomorphic-like properties in a non-associative setting. In this paper, first we present a generalization of the Bohr inequality, and improved versions of the Bohr inequality for slice regular functions over the largest alternative division algebras of octonions $\mathbb{O}$. Moreover, we provide a refined version of the Bohr inequality for slice regular functions $f$ on $\mathbb{B}$ such that $ {\rm Re}(f(x)) \leq 1 $ for all $x \in \mathbb{B}$. All the results are shown to be sharp.
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 $\mathcal{F}(c)$.
In this paper, we present several necessary and sufficient conditions for a logharmonic mapping to be normal i.e., we establish Marty's criterion, Zalcman-Pang lemma and the Lohwater-Pommerenke theorem for logharmonic mappings, along with an application of the Zalcman-Pang lemma.
It is crucial to explore the sharp bounds of logarithmic coefficients and the Hankel determinant involving logarithmic coefficients as part of coefficient problems in various function classes. Our primary objective in this study is to determine the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients of bounded analytic functions associated with a bean-shaped domain in the class $\mathcal{BT_\mathfrak{B}}$. For this class, we also establish the sharp bounds for the second Hankel determinant involving logarithmic coefficients as well as logarithmic inverse coefficients. In addition, we establish sharp bounds for the generalized Zalcman conjecture inequality and the moduli differences of logarithmic coefficients for the class $\mathcal{BT_\mathfrak{B}}$.
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.
It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ of the form \begin{align*} f(z)=z+\sum_{n=2}^{\infty}a_nz^n\; \mbox{for}\; z\in\mathbb{D} \end{align*} and suppose that \begin{align*} F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}γ_{n}(f)z^n, \;\; z\in\mathbb{D},\;\;\log 1:=0, \end{align*} where $ γ_{n}(f) $ is the logarithmic coefficients. The second Hankel determinant of logarithmic coefficients $H_{2,1}(F_{f}/2)$ is defined as: $H_{2,1}(F_{f}/2) :=γ_{1}γ_{3} -γ^2_{2}$, where $γ_1, γ_2,$ and $γ_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we first establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/64$ with logarithmic coefficients for the classes of starlike functions associated with Gregory coefficients. In addition, we establish the sharpness of Fekete-Szeg$ \ddot{o} $ inequality, Zalcman functional and generalized Zalcman functional for the class starlike functions associated with Gregory coefficients.
In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions $f$ on a simply connected domain $Ω$ in $\mathbb{C}$. Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence $φ=\{φ_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions in $[0,1)$ such that the series $\sum_{n=0}^{\infty}φ_n(r)$ converges locally uniformly on the interval $[0,1)$. All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued $ν$-Bloch functions defined in two different simply connected domains, $Ω$ and $Ω_γ$, in $\mathbb{C}$.
In this paper, we first establish refined versions of the Bohr inequalities for the class of holomorphic functions from the unit ball $B_X$ of a complex Banach space $X$ into $\mathbb{C}$. As applications, we will establish refined Bohr inequalities of functional type or of norm type for holomorphic mappings with lacunary series on the unit ball $B_X$ with values in higher dimensional spaces. Next, we obtain the Bohr-Rogosinski inequality for the class of holomorphic functions on $B_X.$ In addition, we establish an improved version of the Bohr inequality for holomorphic functions on $B_X$. All the results are proved to be sharp.
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.
The Bohr radius for an arbitrary class $\mathcal{F}$ of analytic functions of the form $f(z)=\sum_{n=0}^{\infty}a_nz^n$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$ is the largest radius $R_{\mathcal{F}}$ such that every function $f\in\mathcal{F}$ satisfies the inequality \begin{align*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right)=\sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})), \end{align*} for all $|z|=r\leq R_{\mathcal{F}}$ , where $d(0, \partial f(\mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to determine the sharp improved Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relation $zf^{\prime}(z)/f(z)\prec h(z)$ and $f(z)+βzf^{\prime}(z)+γz^2f^{\prime\prime}(z)\prec h(z)$, where $h$ is the Janowski function. We show that improved Bohr radius can be obtained for Janowski functions as root of an equation involving Bessel function of first kind. Analogues results are obtained in this paper for $α$-convex functions and typically real functions, respectively. All obtained results in the paper are sharp and are improved version of [{Bull. Malays. Math. Sci. Soc.} (2021) 44:1771-1785].
In this paper, we first obtain a refined Bohr radius for invariant families of bounded analytic functions on unit disk $ \mathbb{D} $. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, in a simply connected domain \begin{align*} Ω_γ=\biggl\{z\in\mathbb{C}: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ}\;\mbox{for}\; 0\leq γ<1\biggr\}, \end{align*} where $ Ω_0=\mathbb{D} $. These results generalize some existing results. We also show that a better estimate can be obtained in radius and inequality can be shown sharp for Laplace transform of $ f $.
In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $Ω_γ,$ where $0\leq γ<1$ with a sequence $\{φ_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^{\infty}φ_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}(Ω_γ)$, where \begin{align*} Ω_γ:=\biggl\{z\in \mathbb{C}: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ}\biggr\}. \end{align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on $ Ω_γ $. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of $ K $-quasiconformal, sense-preserving harmonic maps of the form $f=h+\overline{g}$ in $Ω_γ.$
In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks $\mathbb{D}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0, 1, 2, 3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{φ_n(r)\}_{n=0}^{\infty}$ of continuous functions of $r$ in $[0, 1)$ and give some applications.
Let $ \mathcal{H}(Ω) $ be the class of complex-valued functions harmonic in $ Ω\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(Ω)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $ r_f\in (0, 1) $ such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where $ d\left(f(0), \partialΩ\right) $ is the Euclidean distance between $ f(0) $ and the boundary of $ Ω:=f(\mathbb{D}) $. The largest such radius $ r_f $ is called the Bohr radius and the inequality $ M_f(r)\leq d\left(f(0), \partialΩ\right) $ is called the Bohr inequality for the class $ \mathcal{H}(Ω) $. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp.
In this paper, a significant improvement has been achieved in the classical Bohr's inequality for the class $ \mathcal{B} $ of analytic self maps defined on the unit disk $ \mathbb{D} $. More precisely, we generalize and improve several Bohr-type inequalities by combining appropriate improved and refined versions of the classical Bohr's inequality with some methods concerning the area measure of bounded analytic functions in $ \mathcal{B} $. In addition, we obtain Bohr-type and Bohr-Rogosinski-type inequalities for the subordination class and also for the class of $ K $-quasiconformal harmonic mappings. All the results are proved to be sharp.
In this paper, we study Bohr's inequality and refined versions of Bohr-Rogosinski inequalities involving Schwarz functions. Moreover, we establish a version of multidimensional analogue of Bohr inequality and Bohr-Rogosinski inequalities involving Schwarz functions. Finally, we establish a multidimensional analogue of the refined version of Bohr inequalities with the initial coefficient being zero. All the results are proved to be sharp.