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Molla Basir Ahamed

Publications and source records attributed to Molla Basir Ahamed.

At least 73 records · Page 4Linked to original sources
arXiv2023

Generalized weighted composition operators on Hardy space $H^2(\mathbb{D}^n)$

In this paper, we explore the complex symmetrical characteristics of weighted composition operators $W_{u, v}$ and weighted composition-differentiation operators $W_{u, v, k_1, k_2, \ldots, k_n}$ on the Hardy space $H^2(\mathbb{D}^n)$ over the Polydisk $\mathbb{D}^n$, with respect to the standard conjugation $\mathcal{J}$. We specify explicit conditions that confirm the Hermitian characteristics of the operator $W_{u, v, k_1, k_2, \ldots, k_n}$ and describe the conditions necessary for it to exhibit normal behavior. Additionally, we identify the kernels of the generalized weighted composition-differentiation operators and their corresponding adjoint operators.

math.FA↗
arXiv2023

Complex symmetric weighted composition operators on the space $\mathcal{H}^2_2(\mathbb{D})$

In this paper, we introduce a new norm for $\mathcal{S}^2(\mathbb{D})$, encompassing functions whose first and second derivatives belong to both the Hardy space $\mathcal{H}^2(\mathbb{D})$ and the classical Bergman space $\mathcal{A}^2(\mathbb{D})$. Moreover, we present some basic properties of the space $\mathcal{H}^2_2(\mathbb{D})$ and subsequently establish conditions for symbols $ϕ$ and $Ψ$ to provide $W_{Ψ, ϕ}$ complex symmetric, employing a unique conjugation $\mathcal{J}$.

math.FA↗
arXiv2023

Operator-valued analogues of multidimensional Bohr-Rogosinski inequalities

In this article, we first establish operator-valued analogues of multidimensional refined Bohr inequality. Then we establish operator-valued analogues of multidimensional improved Bohr inequality with a certain power of the norm of the initial coefficient. Finally, we establish operator-valued analogues of a multidimensional sharp version of Bohr inequality with the initial coefficient being replaced by the norm value of the function. In addition, we establish operator-valued analogues of multidimensional improved Bohr inequality using the quantity $S_r/π$. All the results prove to be sharp.

math.CV↗
arXiv2023

Generalized Bohr inequalities for certain classes of functions and their applications

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $ f\in\mathcal{B} $, then the associated majorant series $ M_f(r):=\sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ holds for $ |z|=r\leq 1/3 $ and the constant $ 1/3 $ cannot be improved. Bohr's original theorem and its subsequent generalizations remain active fields of study, driving investigations in a wide range of function spaces. In this paper, first we establish a generalized Bohr inequality for the class $\mathcal{B}$ by allowing a sequence $\{φ_n(r)\}_{n=0}^{\infty}$ of non-negative continuous functions on $[0, 1)$ in the place of $\{r^n\}_{n=0}^{\infty}$ of the majorant series $M_f(r)$ introducing a weighted sequence of non-negative continuous functions $\{Φ_n(r)\}_{n=0}^{\infty}$ on $[0, 1)$. Secondly, as a generalization, we obtain a refined version of the Bohr inequality for a certain class $\tilde{G}^0_{\mathcal{H}}(β)$ of harmonic mappings. All the results are proved to be sharp.

math.CV↗
arXiv2023

Generalized weighted composition-differentiation operators on weighted Bergman spaces

Let $ \mathcal{H}(\mathbb{D}) $ be the class of all holomorphic functions in the unit disk $ \mathbb{D} $. We aim to explore the complex symmetry exhibited by generalized weighted composition-differentiation operators, denoted as $L_{n, ψ, ϕ}$ and is defined by \begin{align*} L_{n, ψ, ϕ}:=\sum_{k=1}^{n}c_kD_{k, ψ_k, ϕ},\; \mbox{where }\; c_k\in\mathbb{C}\; \mbox{for}\; k=1, 2, \ldots, n, \end{align*} where $ D_{k, ψ, ϕ}f(z):=ψ(z)f^{(k)}(ϕ(z)),\; f\in \mathcal{A}^2_α(\mathbb{D}), $ in the reproducing kernel Hilbert space, labeled as $\mathcal{A}^2_α(\mathbb{D})$, which encompasses analytic functions defined on the unit disk $\mathbb{D}$. By deriving a condition that is both necessary and sufficient, we provide insights into the $ C_{μ, η} $-symmetry exhibited by $L_{n, ψ, ϕ}$. The explicit conditions for which the operator T is Hermitian and normal are obtained through our investigation. Additionally, we conduct an in-depth analysis of the spectral properties of $ L_{n, ψ, ϕ} $ under the assumption of $ C_{μ, η} $-symmetry and thoroughly examine the kernel of the adjoint operator of $L_{n, ψ, ϕ}$.

math.CV↗
arXiv2023

Hankel and Toeplitz determinants of logarithmic coefficients of Inverse functions for certain classes of univalent functions

The Hankel and Toeplitz determinants $H_{2,1}(F_{f^{-1}}/2)$ and $T_{2,1}(F_{f^{-1}}/2)$ are defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix} \;\;\mbox{and} \;\; T_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_1 \end{vmatrix} \end{align*} where $Γ_1, Γ_2,$ and $Γ_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 1/4$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$, $|T_{2,1}(F_{f^{-1}}/2)|\leq 5/16$ and $|T_{2,1}(F_{f^{-1}}/2)|\leq 145/2304$ for the logarithmic coefficients of inverse functions for the classes starlike functions and convex functions with respect to symmetric points. In addition, our findings are substantiated further through the incorporation of illustrative examples, which support the strict inequality and lend credence to our conclusions.

math.CV↗
arXiv2023

Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix}=Γ_1Γ_3-Γ^2_2, \end{align*} where $Γ_1, Γ_2,$ and $Γ_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$ for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order $1/2$, respectively.

math.CV↗
arXiv2023

Entire Solutions for quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) Δ_cf(z) + b [Δ_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, ω, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ Δ_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i<j\leq n $ and their corollaries of the paper are an extension of the results of Xu \emph{et al. } for trinomial equation with arbitrary coefficient in $ \mathbb{C}^2 $. Moreover, examples are exhibited to validate the conclusion of the main results.

math.CV↗
arXiv2023

Characterization of solutions of refined Fermat-type functional equations in $ \mathbb{C}^n $

The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. In fact, we investigate the existence and forms of the transcendental solutions of non-linear quadratic trinomial equations in $\mathbb{C}^n$. As a consequence of our result, we show that solutions of binomial equations in $\mathbb{C}^n$ can be explored and this exploration broaden the scope of the study of functional equations in $ \mathbb{C}^n $. The results we obtained are improvements over certain recent findings, noted as remarks. In addition, some examples relevant to the content of the paper have been exhibited in support of the validation of each results. Further, we discuss situations when solutions of such equations does not exist.

math.CV↗
arXiv2023

Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune

The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} γ_1 & γ_2 γ_2 & γ_3 \end{vmatrix}, \end{align*} 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 establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/16$ and $|H_{2,1}(F_{f}/2)| \leq 23/3264$ for the logarithmic coefficients of starlike and convex functions associated with lune.

math.CV↗
arXiv2023

Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions

The Hankel determinant $H_{2,2}(F_{f}/2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} γ_2 & γ_3 γ_3 & γ_4 \end{vmatrix}, \end{align*} where $γ_2, γ_3,$ and $γ_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,2}(F_{f}/2)|\leq (1272 + 113\sqrt{678})/32856$ and $|H_{2,2}(F_{f}/2)| \leq 13/1080$ for the logarithmic coefficients of starlike and convex functions with respect to symmetric points. Moreover, we provide examples that demonstrate the strict inequality holds.

math.CV↗
arXiv2023

Refined Bohr inequality for functions in $\mathbb{C}^n$ and in complex Banach spaces

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $\mathbb{C}^n$ under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain $ G $ of a complex Banach space $ X $ and take values from the unit disk $ \mathbb{D} $. Furthermore, as a consequence of one of this results, we obtain a refined version of the Bohr-type inequalities for harmonic functions $ f=h+\bar{g} $ defined on a balanced domain $ G\subset X $. All the results are proved to be sharp.

math.CV↗
arXiv2022

The sharp refined Bohr-Rogosinski inequalities for certain classes of harmonic mappings

A class $ \mathcal{F} $ consisting of analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disc $ \mathbb{D}=\{z\in\mathbb{C}:|z|<1\} $ satisfies a Bohr phenomenon if there exists an $ r_f>0 $ such that \begin{equation*} I_f(r):=\sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial \mathbb{D}\right) \end{equation*} for every function $ f\in\mathcal{F} $, and $ |z|=r\leq r_f $. The largest radius $ r_f $ is the Bohr radius and the inequality $ I_f(r)\leq{d}\left(f(0),\partial \mathbb{D}\right) $ is Bohr inequality for the class $ \mathcal{F} $, where `$ d $' is the Euclidean distance. If there exists a positive real number $ r_0 $ such that $ I_f(r)\leq {d}\left(f(0),\partial \mathbb{D}\right) $ holds for every element of the class $ \mathcal{F} $ for $ 0\leq r r_0 $, then we say that $ r_0 $ is sharp bound for the inequality w.r.t. the class $ \mathcal{F} $. In this paper, we prove sharp refinement of the Bohr-Rogosinski inequality for certain classes of harmonic mappings.

math.CV↗
arXiv2022

Improved Bohr inequalities for certain classes of harmonic mappings

The Bohr radius for the class of harmonic functions of the form $ f(z)=h+\overline{g} $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $, where $ h(z)=\sum_{n=0}^{\infty}a_nz^n $ and $ g(z)=\sum_{n=1}^{\infty}b_nz^n $ is to find the largest radius $ r_f $, $ 0<r_f<1 $ such that \begin{equation*} \sum_{n=1}^{\infty}\left(|a_n|+|b_n|\right)|z|^n\leq d(f(0),\partial f(\mathbb{D})) \end{equation*} holds for $ |z|=r\leq r_f $, where $ d(f(0),\partial f(\mathbb{D})) $ is the Euclidean distance between $ f(0) $ and the boundary of $ f(\mathbb{D}) $. In this paper, we prove two-type of improved versions of the Bohr inequalities, one for a certain class of harmonic and univalent functions and the other for stable harmonic mappings. It is observed in the paper that to obtain sharp Bohr inequalities it is enough to consider any non-negative real coefficients of the quantity $ S_r/π$. As a consequence of the main result, we prove corollaries showing the precise value of the sharp Bohr radius.

math.CV↗
arXiv2022

Transcendental entire solutions of several general quadratic type PDEs and PDDEs in $ \mathbb{C}^2 $

The functional equations $ f^2+g^2=1 $ and $ f^2+2αfg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. Utilizing Nevanlinna's theory for several complex variables, in this paper, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form $ af^2+2αfg+b g^2+2βf+2γg+C=0 $ in $ \mathbb{C}^2 $. Consequently, we obtain certain corollaries of the main results of this paper concerning binomial equations which generalize many results in [\textit{Rocky Mountain J. Math.} \textbf{51}(6) (2021), 2217-2235] in the sense of arbitrary coefficients.

math.CV↗
arXiv2021

Class of meromorphic functions partially shared values with their differences or shifts

Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function $ f $ share $ s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $ E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". With the idea of partially shared values, in this paper, we prove some uniqueness results between non-constant meromorphic functions and their shifts or generalized differences. We exhibit some examples to show that the result of {Charak \emph{et al.}} \cite{Cha & Kor & Kum-2016} is not true for $k=2 $ or $ k=3 $. We find some gaps in proof of the result of {Lin} \emph{et al.} \cite{Lin & Lin & Wu}, and we not only correct them but also generalize their result in a more convenient way. A number of examples have been exhibited to validate a certain claim of the main results of this paper and also to show that some of the conditions are sharp. In the end, we have posed some open questions for further investigation of the main result of the paper.

math.CV↗
arXiv2021

Bohr-Rogosinski and improved Bohr type inequalities for certain fully starlike harmonic mappings

The classical Bohr inequality states that if $ f $ is an analytic function with the power series representation $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $ such that $ |f(z)|\leq 1 $ for all $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\; \text{for}\;\; |z|=r\leq\frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ r_0=1/3 $ is known as Bohr radius and the inequality $ \sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ is known as Bohr inequality. Let $ \mathcal{H} $ be the class of complex-valued harmonic mappings $ f=h+\bar{g}$ defined in the unit disk $ \mathbb{D} $, where $ h $ and $ g $ are analytic functions in $ \mathbb{D} $ with the normalization $ h(0)=0=h^{\prime}(0)-1 $ and $ g(0)=0 $. Let $ \mathcal{H}_{0}=\{f=h+\bar{g}\in\mathcal{H} : g^{\prime}(0)=0\}. $ Let $ \mathcal{P}^{0}_{\mathcal{H}}(M) :=\{f=h+\overline{g} \in \mathcal{H}_{0}: \real (zh^{\prime\prime}(z))> -M+|zg^{\prime\prime}(z)|,\; z \in \mathbb{D},\; M>0\} $. Functions in the class $ \mathcal{P}^{0}_{\mathcal{H}}(M) $ are called fully starlike univalent functions for $ 0<M<1/\log 4 $. In this paper, we obtain the sharp Bohr-Rogosinski type inequality and improved Bohr inequality and the corresponding Bohr radius for the class $ \mathcal{P}_{\mathcal{H}}^{0}(M) $.

math.CV↗