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Vasudevarao Allu

Publications and source records attributed to Vasudevarao Allu.

At least 55 records · Page 3Linked to original sources
arXiv2024

Arithmetic Bohr radius of bounded linear operators

In this paper, we investigate the arithmetic Bohr radius of bounded linear operators between arbitrary complex Banach spaces. We establish the close connection between the classical Bohr radius and the arithmetic Bohr radius of bounded linear operators. Further, we study the asymptotic estimates of arithmetic Bohr radius for identity operator on infinite dimensional complex Banach spaces. Finally, we obtain the correct asymptotic behavior of Bohr radii of operators between sequence spaces.

math.FA↗
arXiv2024

Landau type theorem for $α$-harmonic mappings

In this paper, we first obtain an estimate of the coefficients for $α$-harmonic mappings. By applying these coefficient estimates, we prove the Landau type theorem for $α$-harmonic mappings defined on the unit disc $\ID$.

math.CV↗
arXiv2024

On the generalized Zalcman conjecture

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the generalized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In the same paper, Ma \cite{Ma-1999} asked for what positive real values of $λ$ does the following inequality hold? \begin{equation}\label{conjecture} |λa_na_m-a_{n+m-1}|\le λnm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge3). \end{equation} Clearly equality holds for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper, we prove the inequality (\ref{conjecture}) for $λ=3, n=2, m=3$. Further, we provide a geometric condition on extremal function maximizing (\ref{conjecture}) for $λ=2,n=2, m=3$.

math.CV↗
arXiv2024

Moduli difference of inverse logarithmic coefficients of univalent functions

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some disk $|w|\le r_0(f)$. The inverse logarithmic coefficients $Γ_n$, $n \in \mathbb{N}$, of $f$ are defined by the equation $ \log(F(w)/w)=2\sum_{n=1}^{\infty}Γ_{n}w^{n},\,|w|<1/4.$ In this paper, we find the sharp upper and lower bounds for moduli difference of second and first inverse logarithmic coefficients, {\em i.e.,} $|Γ_2|-|Γ_1|$ for functions in class $\mathcal{S}$ and for functions in some important subclasses of univalent functions.

math.CV↗
arXiv2023

On invariance of John domains under quasisymmetric mappings

In this paper, we prove that if a homeomorphism is quasisymmetric relative to the boundary of the domain then it maps a length John domain to a diameter John domain. Moreover, we prove a necessary and sufficient condition for a diameter John domain to be length John and thereby prove that if $f:G\rightarrow G'$ is $(M, C)-$CQH map, where $G$ is a John domain, and the map extends to the boundary such that the extension is $η-$QS relative to $δG$ then $G'$ is a John domain. In addition, we characterize distance John domains using the weak minimizing property.

math.CV↗
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

Composition-Differentiation Operator on Weighted Bergman Spaces

In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, ψ, ϕ}$ on weighted Bergman spaces $\mathcal{A}^2_α$ with respect to the conjugation $C_{μ, η}$ for $μ, η\in \{z\in \mathbb{C}:|z|=1\}$. We obtain explicit conditions for which the operator $D_{n, ψ, ϕ}$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces.

math.CV↗
arXiv2022

Landau-Bloch type theorem for elliptic and $K$-quasiregular harmonic mappings

In this paper, we establish an improved coefficient bounds for quasiregular and elliptic harmonic mappings and using these bounds we establish Landau-Bloch type theorem for $(K,K')$-elliptic and K-quasiregular harmonic mappings in plane. Furthermore, we prove the coefficient estimates for $K$-quasiconformal harmonic self maps defined on the unit disk $\mathbb{D}$.

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

Region of variability for certain subclass of univalent functions

Let $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ be the unit disk. For $0<α<1$, let $f_α(z)=z/(1-z^α)$ for $z \in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_α$ which satisfy $\Re \left(1+zf"_α(z)/f'_α(z)\right) > β$ for $0<β<1$. In this paper, we determine the region of variability of $\log f'_α(z_0)$ for fixed $z_{0} \in \mathbb{D}$ when $f$ varies over the class ${\mathcal F}(λ):=\{f_α \in \mathcal{F}: f_α(0)=0, f'_α(0)=1 \, \mbox{and} \, f"_α(0)=2λ(1-β) \,\,\, \mbox{for} \,\, 0\leq λ\leq 1\}$.

math.CV↗
arXiv2022

Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, Ma proposed the generalized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper using the properties of holomorphic motion and Krushkal's Surgery Lemma \cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$, $m=3$ and $n=2$, $m=4$.

math.CV↗
arXiv2022

Bohr and Rogosinski inequalities for operator valued holomorphic functions

For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each $N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly $\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order $N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q (μ)$ for $1\leq p<q<\infty$ or $1\leq q \leq p <2$. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk $\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$ denotes the space of all bounded linear operator on a complex Hilbert space $\mathcal{H}$.

math.FA↗
arXiv2022

Support points of some classes of analytic and univalent functions

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$\left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D},$$ and $\mathscr{G}$ denote the class of functions $f\in \mathcal{A}$ satisfying $${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \quad\mbox{ for } z\in\mathbb{D}.$$

math.CV↗
arXiv2022

Bohr-Rogosinski type inequalities for concave univalent functions

In this paper, we generalize and investigate Bohr-Rogosinski's inequalities and the Bohr-Rogosinski phenomenon for the subfamilies of univalent (i.e., one-to-one) functions defined on unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$ which maps to the concave domain, i.e., the domain whose complement is a convex set. All the results are proved to be sharp.

math.CV↗
arXiv2022

On Bloch norm and Bohr phenomenon for harmonic Bloch functions on simply connected domains

In this article, we introduce the class $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch-type mappings on $Ω$ as a generalization of the class $\mathcal{B}_{\mathcal{H},Ω}(α)$ of harmonic $α$-Bloch mappings on $Ω$, where $Ω$ is arbitrary proper simply connected domain in the complex plane. We study several interesting properties of the classes $\mathcal{B}_{\mathcal{H},Ω}(α)$ and $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ on arbitrary proper simply connected domain $Ω$ and on the shifted disk $Ω_γ$ containing $\mathbb{D}$, where $$ Ω_γ:=\bigg\{z\in\mathbb{C} : \bigg|z+\fracγ{1-γ}\bigg|<\frac{1}{1-γ}\bigg\ $$ and $0 \leq γ<1$. We establish the Landau's theorem for the harmonic Bloch space $\mathcal{B}_{\mathcal{H},Ω_γ}(α)$ on the shifted disk $Ω_γ$. For $f \in \mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$) of the form $f(z)=h(z) + \overline{g(z)}=\sum_{n=0}^{\infty}a_nz^n + \overline{\sum_{n=1}^{\infty}b_nz^n}$ in $\mathbb{D}$ with Bloch norm $||f||_{\mathcal{H},Ω, α} \leq 1$ (respectively $||f||^{*}_{\mathcal{H},Ω, α} \leq 1$), we define the Bloch-Bohr radius for the space $\mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$) to be the largest radius $r_{Ω,f} \in (0,1)$ such that $\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq 1$ for $r \leq r_{Ω, α}$ and for all $f \in \mathcal{B}_{\mathcal{H},Ω}(α)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$). We investigate Bloch-Bohr radius for the classes $\mathcal{B}_{\mathcal{H},Ω}(α)$ and $\mathcal{B}^{*}_{\mathcal{H},Ω}(α)$ on simply connected domain $Ω$ containing $\mathbb{D}$.

math.CV↗