arXiv subjects
Publications and source records attributed to Vasudevarao Allu.
In this article, we study the Bohr type inequality for {C}esáro operator and {B}ernardi integral operator acting on the space of analytic functions defined on a simply connected domain containing the unit disk $\mathbb{D}$.
We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second Hankel determinant of Logarithmic coefficients for some subclasses of analytic functions.
Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. In this paper, we first establish several sharp improved and refined versions of the Bohr's inequality for the functions in the class $H^{\infty}(\mathbb{D},\mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}:=\{z \in \mathbb{C}:|z|<1\}$ into $\mathcal{B}(\mathcal{H})$. For the complete circular domain $Q \subset \mathbb{C}^n$, we prove the multidimensional analogues of the operator valued Bohr's inequality established by G. Popescu [Adv. Math. 347 (2019), 1002-1053]. Finally, we establish the multidimensional analogues of several improved Bohr's inequalities for operator valued functions in $Q$.
Let $H^{\infty}(Ω,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $Ω$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ into a complex Banach space $X$ with $\norm{f}_{H^{\infty}(Ω,X)} \leq 1$. Let $ϕ=\{ϕ_{n}(r)\}_{n=0}^{\infty}$ with $ϕ_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} ϕ_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q<\infty$, we denote \begin{equation*} R_{p,q,ϕ}(f,Ω,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p ϕ_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}ϕ_{n}(r)\right)^q \leq ϕ_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with $ϕ$ by $$R_{p,q,ϕ}(Ω,X)=\inf \left\{R_{p,q,ϕ}(f,Ω,X): \norm{f}_{H^{\infty}(Ω,X)} \leq 1\right\}.$$ In this article, we extensively study the Bohr radius $R_{p,q,ϕ}(Ω,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.
In this article, we study the Bohr operator for the operator valued subordination class $S(f)$ consisting of holomorphic functions subordinate to $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$, where $f:\mathbb{D} \rightarrow \mathcal{B}(\mathcal{H})$ is holomorphic and $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal{H}$. We establish several subordination results, which can be viewed as the analogues of a couple of interesting subordination results from scalar valued settings. We also obtain a von Neumann-type inequality for the class of self-analytic mappings of the unit disk $\mathbb{D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator valued polyanalytic functions in certain proper simply connected domains in $\mathbb{C}$. We obtain Bohr radius for the operator valued polyanalytic functions of the form $F(z)= \sum_{l=0}^{p-1} \overline{z}^l \, f_{l}(z) $, where $f_{0}$ is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk $\mathbb{D}$.
We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some $|w|\le r_0(f)$. In this paper, we find the sharp bounds of $\big | |A_{n+1}|-|A_n|\big |$, for $n=1,\,2$, for some subclasses of univalent functions.
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) $.
Bohr phenomenon for analytic functions $ f $ where $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, first introduced by Harald Bohr in $ 1914 $, deals with finding the largest radius $ r_f $, $ 0<r_f<1 $, such that the inequality $ \sum_{n=0}^{\infty}|a_nz^n|<1 $ holds whenever $ |f(z)|<1 $ holds in the unit disk $ \mathbb{D}=\{z\in\mathbb{C} : |z|<1\} $. The Bohr phenomenon for the harmonic functions of the form $ f(z)=h+\overline{g} $, 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|\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 several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.
Let $ \mathcal{H}(\mathbb{D}) $ be the linear space of analytic functions on the unit disk $ \mathbb{D}=\{z\in\mathbb{C}: |z|<1\} $ and let $ \mathcal{B}=\{w\in \mathcal{H}(\mathbb{D}: |w(z)|<1)\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ is the best possible. The constant $ 1/3 $ is known as Bohr radius. A function $ f : \mathbb{D}\rightarrow\mathbb{C} $ is said to be log-harmonic if there is a $ w\in\mathcal{B} $ such that $ f $ is a non-constant solution of the non-linear elliptic partial differential equation \begin{equation*} \bar{f}_{\bar{z}}(z)/\bar{f}(z)=w(z)f_{z}(z)/f(z). \end{equation*} The class of log-harmonic mappings is denoted by $ \mathcal{S}_{LH} $. The set of all starlike log-harmonic mapping is defined by \begin{equation*} \mathcal{ST}_{LH}=\bigg\{f\in\mathcal{S}_{LH}:\frac{\partial}{\partialθ}{\rm Arg}(f(e^{iθ}))={\rm Re}\left(\frac{zf_{z}-\bar{z}f_{\bar{z}}}{f}\right)>0\;\; \mbox{in}\;\; \mathbb{D}\bigg\}. \end{equation*} In this paper, we study several improved Bohr radius for the class $ \mathcal{ST}^{0}_{LH} $, a subclass of $ \mathcal{ST}_{LH} $, consisting of functions $ f\in\mathcal{ST}_{LH} $ which map the unit disk $ \mathbb{D} $ onto a starlike domain (with respect to the origin).
Let $ϕ$ be analytic and univalent ({\it i.e.,} one-to-one) in $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $ϕ(\mathbb{D})$ has positive real part, is symmetric with respect to the real axis, starlike with respect to $ϕ(0)=1,$ and $ϕ' (0)>0$. A function $f \in \mathcal{C}(ϕ)$ if $1+ zf''(z)/f'(z) \prec ϕ(z),$ and $f\in \mathcal{C}_{c}(ϕ)$ if $2(zf'(z))'/(f(z)+\overline{f(\bar{z})})' \prec ϕ(z)$ for $ z\in \mathbb{D}$. In this article, we consider the classes $\mathcal{HC}(ϕ)$ and $\mathcal{HC}_{c}(ϕ)$ consisting of harmonic mappings $f=h+\overline{g}$ of the form $$ h(z)=z+ \sum \limits_{n=2}^{\infty} a_{n}z^{n} \quad \mbox{and} \quad g(z)=\sum \limits_{n=2}^{\infty} b_{n}z^{n} $$ in the unit disk $\mathbb{D}$, where $h$ belongs to $\mathcal{C}(ϕ)$ and $\mathcal{C}_{c}(ϕ)$ respectively, with the dilation $g'(z)=αz h'(z)$ and $|α|<1$. Using the Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we find the radius $R_{f}<1$ such that Bohr inequality $$ |z|+\sum_{n=2}^{\infty} (|a_{n}|+|b_{n}|)|z|^{n} \leq d(f(0),\partial f(\mathbb{D})) $$ holds for $|z|=r\leq R_{f}$ for the classes $\mathcal{HC}(ϕ)$ and $\mathcal{HC}_{c}(ϕ)$ . As a consequence of these results, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes.
Let $ \mathcal{H} $ be the class of complex-valued harmonic mappings $ f=h+\bar{g}$ defined in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $, 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\}. $ Ghosh and Vasudevrao \cite{Ghosh-Vasudevarao-BAMS-2020} have studied the following interesting harmonic univalent class $ \mathcal{P}^{0}_{\mathcal{H}}(M) $ which is defined by $$\mathcal{P}^{0}_{\mathcal{H}}(M) :=\{f=h+\overline{g} \in \mathcal{H}_{0}: \mathrm{Re} (zh^{\prime\prime}(z))> -M+|zg^{\prime\prime}(z)|,\; z \in \mathbb{D}\; \mbox{and}\;\; M>0\}. $$ In this paper, we obtain the sharp Bohr-Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class $ \mathcal{P}_{\mathcal{H}}^{0}(M) $.
Let $ \mathcal{H} $ be the class of harmonic functions $ f=h+\bar{g} $ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C} : |z|<1\}$, where $ h $ and $ g $ are analytic in $ \mathbb{D} $. Let $$\mathcal{P}_{\mathcal{H}}^{0}(α)=\{f=h+\overline{g} \in \mathcal{H} : \real (h^{\prime}(z)-α)>|g^{\prime}(z)|\; \mbox{with}\; 0\leqα<1,\; g^{\prime}(0)=0,\; z \in \mathbb{D}\} $$ be the class of close-to-convex mappings defined by Li and Ponnusamy \cite{Injectivity section}. In this paper, we obtain the sharp Bohr-Rogosinski radius, improved Bohr radius and refined Bohr radius for the class $ \mathcal{P}_{\mathcal{H}}^{0}(α) $.
Let $ \mathcal{H}(\mathbb{D}) $ be the class of analytic functions in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ 1/3 $ is known as Bohr radius. In this paper, we study Bohr phenomenon for analytic as well as harmonic mappings on simply connected domains. We prove several sharp results on improved Bohr radius for analytic functions as well as for harmonic mappings on simply connected domains.
In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain \begin{equation*} Ω_γ=\bigg\{z\in\mathbb{C} : \bigg|z+\fracγ{1-γ}\bigg|<\frac{1}{1-γ}\bigg\}\;\; \text{for}\;\; 0\leq γ<1. \end{equation*} We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in $ Ω_γ $, and obtain several sharp results.
We say that a class $\mathcal{B}$ of analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_{f}<1$, the following inequality $$ \sum\limits_{n=1}^{\infty} |a_{n}z^{n}| \leq d(f(0),\partial f(\mathbb{D}) ) $$ holds for $|z|=r\leq R_{f}$ and for all functions $f \in \mathcal{B}$. The largest radius $R_{f}$ is called Bohr radius for the class $\mathcal{B}$. In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes $\mathcal{S}_{c}^{*}(ϕ),\,\mathcal{C}_{c}(ϕ),\, \mathcal{C}_{s}^{*}(ϕ),\, \mathcal{K}_{s}(ϕ)$. Using Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we obtain some radius $R_{f}$ such that Bohr phenomenon for these classes holds for $|z|=r\leq R_{f}$. Generally, in this case $R_{f}$ need not be sharp, but we show that under some additional conditions on $ϕ$, the radius $R_{f}$ becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes.
We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_Ω^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset Ω, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when $Ω\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. In the present article, we first show that in the case $n=0$, $j=-1$ and $c=0$, the result obtained in \cite{Ali-Vasudevarao-Yanagihara-2018} still holds when one assumes only that $Ω$ is starlike with respect to $P(0)$. Let $\mathcal{CV}(Ω)$ be the class of analytic functions $f$ in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in Ω$. As applications we determine variability regions of $\log f'(z_0)$ when $f$ ranges over $\mathcal{CV}(Ω)$ with or without the conditions $f''(0)= λ$ and $f'''(0)= μ$. Here $λ$ and $μ$ are arbitrarily preassigned values. By choosing particular $Ω$, we obtain the precise variability regions of $\log f'(z_0)$ for other well-known subclasses of analytic and univalent functions.
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U}(λ)$. We also obtain the estimate of the Fekete-Szegö functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in $\mathcal{U}(λ)$. Further, for the class of meromorphic functions which are defined in $Δ:=\{ζ\in\mathbb{\widehat{C}}:|ζ|>1\}$ and associated with the class $\mathcal{U}(λ)$, we obtain a sufficient condition for a function $g$ to be an extreme point of this class.
We say that a class $\mathcal{F}$ consisting of analytic functions $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if there exists $r_{f} \in (0,1)$ such that $$ \sum_{n=1}^{\infty} |a_{n}z^{n}|\leq d(f(0),\partial f(\mathbb{D})) $$ for every function $f \in \mathcal{F}$ and $|z|=r\leq r_{f}$, where $d$ is the Euclidean distance. The largest radius $r_{f}$ is the Bohr radius for the class $\mathcal{F}$. In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point.