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In this paper, we establish three Landau-type theorems for certain bounded poly-analytic functions, which generalize the corresponding result for bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1): 2024, 152-165]. Further, we prove three bi-Lipschitz theorems for these subclasses of poly-analytic functions., Comment: 15 pages

For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary can be uniformized by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification of the Gromov boundary of $X$ with the metric boundary of the deformed space., Comment: More details for some proofs have been added

In this paper, we study a more general version of multidimensional Bohr radii for the holomorphic functions defined on unit ball of $\ell^n_q\,\,(1\leq q\leq \infty)$ spaces with values in arbitrary complex Banach spaces. More precisely, we study the multidimensional Bohr radii for bounded linear operators between complex Banach spaces, primarily motivated by the work of A. Defant, M. Maestre, and U. Schwarting [Adv. Math. 231 (2012), pp. 2837--2857]. We obtain the exact asymptotic estimates of multidimensional Bohr radius for both finite and infinite dimensional Banach spaces. As an application, we find the lower bound of arithmetic Bohr radius., Comment: 19 pp; Some typos has been corrected

In this paper, we first obtain an estimate of the coefficients for $\alpha$-harmonic mappings. By applying these coefficient estimates, we prove the Landau type theorem for $\alpha$-harmonic mappings defined on the unit disc $\ID$., Comment: 12

In this paper, we introduce the concept of quasihyperbolically visible spaces. As a tool, we study the connection between the Gromov boundary and the metric boundary., Comment: 12 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2306.03815

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 $\Gamma_n$, $n \in \mathbb{N}$, of $f$ are defined by the equation $ \log(F(w)/w)=2\sum_{n=1}^{\infty}\Gamma_{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.,} $|\Gamma_2|-|\Gamma_1|$ for functions in class $\mathcal{S}$ and for functions in some important subclasses of univalent functions.

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.

The main aim of this paper is to study the $n$-dimensional Bohr radius for holomophic functions defined on Reinhardt domain in $\mathbb{C}^n$ with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611--2619]. A part of our investigation in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space $\ell^n_{q}\, , 1\leq q\leq \infty$. Further, we determine the exact value of Bohr radius in terms of arithmetric Bohr radius., Comment: 11 Pages

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 $\eta-$QS relative to $\delta G$ then $G'$ is a John domain. In addition, we characterize distance John domains using the weak minimizing property.

In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the strongly starlike and strongly convex functions of order alpha., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2306.15467

The main aim of this paper is to answer certain open questions related to the exact values of multidimensional Bohr radii by using the concept of arithmetic Bohr radius for vector-valued holomorphic functions defined in complete Reinhardt domains in $\mathbb{C}^n$. More precisely, we study the asymptotic estimates of the arithmetic Bohr radius for holomorphic functions in the unit ball of $\ell^n_q$ $(1\leq q\leq \infty)$ spaces with values in arbitrary complex Banach spaces. Many of our results generalize the results obtained by Defant, Maestre, and Prengel [Q. J. Math. 59, (2008), pp. 189--205]., Comment: We have revised some of proof of our results

In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the starlike and convex functions., Comment: 13 Pages

In this article, we initiate the study of visibility property in the context of quasihyperbolic metric. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $\partial_{Euc}\Omega$ be its Euclidean boundary. We say that the pair $\{p,q\}\subset\partial_{Euc}\Omega$ has visible quasihyperbolic geodesic if there exist open neighborhoods $U_p$ of $p$ and $U_q$ of $q$ and a compact set $K_{\{p,q\}}\subset \Omega$ such that for any quasihyperbolic geodesic which joins points in $U_p$ and $U_q$ intersects with $K$. If a domain has visible quasihyperbolic geodesic for every pair of points, we say that it is a visibility domain. The major part of this paper is devoted to provide a rich collection of visibility domains. In this direction we provide a general visibility criteria for a domain to be a visibility domain, and using this we obtain the visibility of Uniform domains, John domains and domains satisfying quasihyperbolic boundary conditions. We also study the visibility of bounded hyperbolic domains with respect to hyperbolic and quasihyperbolic metric. Further, we explore the relation between Gromov hyperbolicity and visibility. As an application of visibility we prove a sufficient condition for the continuous extension of quasihyperbolic isometries and quasi-isometries. Further, motivated by the work of Bharali and Zimmer [{\it Trans. Amer. Math. Soc. (2022)}] and using the end compactification we define the visibility of unbounded domains and study its relationship with Gromov hyperbolicity. Finally, we provide some useful examples of visibility domains which are neither John domain nor domains satisfying quasihyperbolic boundary conditions., Comment: 51 pages, 2 figures. A revised version with significant additions. For instance, visibility of unbounded domains

In this paper, we prove that neargeodesics in Gromov hyperbolic John domains in Banach space are cone arcs. This result gives an improvement of a result of Li [Theorem 1, Int. J. Math. 25 (2014)]., Comment: 14 pages. Revised version accepted for publication in Annales Fennici Mathematici

In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, \psi, \phi}$ on weighted Bergman spaces $\mathcal{A}^2_{\alpha}$ with respect to the conjugation $C_{\mu, \eta}$ for $\mu, \eta \in \{z\in \mathbb{C}:|z|=1\}$. We obtain explicit conditions for which the operator $D_{n, \psi, \phi}$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces., Comment: 14 pages

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 $\mathbb{D}$., Comment: 13 pages

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

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 $${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \quad\mbox{ for } z\in\mathbb{D}.$$, Comment: 10 pages

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 $\lambda$ does the following inequality hold? \begin{equation}\label{conjecture} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -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 $\lambda=3, n=2, m=3$. Further, we provide a geometric condition on extremal function maximizing (\ref{conjecture}) for $\lambda=2,n=2, m=3$., Comment: Revised version, accepted for publication in Annali di Matematica Pura ed Applicata (1923 -), 11 Pages

Let $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ be the unit disk. For $0<\alpha <1$, let $f_{\alpha}(z)=z/(1-z^\alpha)$ for $z \in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_{\alpha}$ which satisfy $\Re \left(1+zf"_{\alpha}(z)/f'_{\alpha}(z)\right) > \beta$ for $0<\beta<1$. In this paper, we determine the region of variability of $\log f'_{\alpha}(z_0)$ for fixed $z_{0} \in \mathbb{D}$ when $f$ varies over the class ${\mathcal F}(\lambda):=\{f_{\alpha} \in \mathcal{F}: f_{\alpha}(0)=0, f'_{\alpha}(0)=1 \, \mbox{and} \, f"_{\alpha}(0)=2\lambda (1-\beta) \,\,\, \mbox{for} \,\, 0\leq \lambda \leq 1\}$., Comment: 19 pages, 10 figures

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 $n=2$, $m=4$., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2209.10595

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.

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 $L^q (\mu)$ for $1\leq p

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 $Q$., Comment: We revise the proof of Lemma 3.1

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 $\mathbb{D}$., Comment: We revise the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in this article. 11 pages

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $\Omega$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ into a complex Banach space $X$ with $\norm{f}_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q<\infty$, we denote \begin{equation*} R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with $\phi$ by $$R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \norm{f}_{H^{\infty}(\Omega,X)} \leq 1\right\}.$$ In this article, we extensively study the Bohr radius $R_{p,q,\phi}(\Omega,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Ces\'{a}ro operator and Bernardi operator., Comment: We revise the proof of Theorem 1.2. This paper contains 23 pages, 12 figures, 6 tables

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.

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.

In this article, we introduce the class $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ of harmonic $\alpha$-Bloch-type mappings on $\Omega$ as a generalization of the class $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ of harmonic $\alpha$-Bloch mappings on $\Omega$, where $\Omega$ is arbitrary proper simply connected domain in the complex plane. We study several interesting properties of the classes $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ and $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ on arbitrary proper simply connected domain $\Omega$ and on the shifted disk $\Omega_{\gamma}$ containing $\mathbb{D}$, where $$ \Omega_{\gamma}:=\bigg\{z\in\mathbb{C} : \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\ $$ and $0 \leq \gamma <1$. We establish the Landau's theorem for the harmonic Bloch space $\mathcal{B}_{\mathcal{H},\Omega _{\gamma}}(\alpha)$ on the shifted disk $\Omega_{\gamma}$. For $f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$) 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 $||f||_{\mathcal{H},\Omega, \alpha} \leq 1$ (respectively $||f||^{*}_{\mathcal{H},\Omega, \alpha} \leq 1$), we define the Bloch-Bohr radius for the space $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$) to be the largest radius $r_{\Omega,f} \in (0,1)$ such that $\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq 1$ for $r \leq r_{\Omega, \alpha}$ and for all $f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ (respectively $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$). We investigate Bloch-Bohr radius for the classes $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ and $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ on simply connected domain $\Omega$ containing $\mathbb{D}$., Comment: We missed to cite the paper of Liu and Ponnusamy [49] at the proper places. We have cited the article [49] at the proper places. 36 pages, 14 figures

In this article, we study the Bohr type inequality for {C}es\'{a}ro operator and {B}ernardi integral operator acting on the space of analytic functions defined on a simply connected domain containing the unit disk $\mathbb{D}$., Comment: 9 pages

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

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

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}{\partial\theta}{\rm Arg}(f(e^{i\theta}))={\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 $ \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) $., Comment: 19 pages, 3 figures. arXiv admin note: text overlap with arXiv:2012.06829

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 $$\mathcal{P}_{\mathcal{H}}^{0}(\alpha)=\{f=h+\overline{g} \in \mathcal{H} : \real (h^{\prime}(z)-\alpha)>|g^{\prime}(z)|\; \mbox{with}\; 0\leq\alpha<1,\; g^{\prime}(0)=0,\; z \in \mathbb{D}\} $$ 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}(\alpha) $., Comment: 26 pages, 21 figures

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., Comment: 17 pages, 2 figures

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain \begin{equation*} \Omega_{\gamma}=\bigg\{z\in\mathbb{C} : \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}\;\; \text{for}\;\; 0\leq \gamma<1. \end{equation*} We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in $ \Omega_{\gamma} $, and obtain several sharp results., Comment: 18 pages

Let $\phi$ be analytic and univalent ({\it i.e.,} one-to-one) in $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $\phi(\mathbb{D})$ has positive real part, is symmetric with respect to the real axis, starlike with respect to $\phi(0)=1,$ and $\phi ' (0)>0$. A function $f \in \mathcal{C}(\phi)$ if $1+ zf''(z)/f'(z) \prec \phi (z),$ and $f\in \mathcal{C}_{c}(\phi)$ if $2(zf'(z))'/(f(z)+\overline{f(\bar{z})})' \prec \phi (z)$ for $ z\in \mathbb{D}$. In this article, we consider the classes $\mathcal{HC}(\phi)$ and $\mathcal{HC}_{c}(\phi)$ 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 $\mathcal{C}(\phi)$ and $\mathcal{C}_{c}(\phi)$ respectively, with the dilation $g'(z)=\alpha z h'(z)$ and $|\alpha|<1$. 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 $\mathcal{HC}(\phi)$ and $\mathcal{HC}_{c}(\phi)$ . As a consequence of these results, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes., Comment: 18 pages, 4 figures

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 $\mathcal{S}_{c}^{*}(\phi),\,\mathcal{C}_{c}(\phi),\, \mathcal{C}_{s}^{*}(\phi),\, \mathcal{K}_{s}(\phi)$. 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 $\phi$, 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., Comment: 18 pages

For $0<\lambda\le 1$, let $\mathcal{U}(\lambda)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<\lambda$ 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 $\mathcal{U}(\lambda)$. We also obtain the estimate of the Fekete-Szeg\"{o} functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in $\mathcal{U}(\lambda)$. Further, for the class of meromorphic functions which are defined in $\Delta:=\{\zeta\in\mathbb{\widehat{C}}:|\zeta|>1\}$ and associated with the class $\mathcal{U}(\lambda)$, we obtain a sufficient condition for a function $g$ to be an extreme point of this class., Comment: 10 pages This paper is with the Journal for the last one year and six months

We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset \Omega, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ 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 $\Omega\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. 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 $\Omega$ is starlike with respect to $P(0)$. Let $\mathcal{CV}(\Omega)$ be the class of analytic functions $f$ in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega}$. As applications we determine variability regions of $\log f'(z_0)$ when $f$ ranges over $\mathcal{CV}(\Omega)$ with or without the conditions $f''(0)= \lambda$ and $f'''(0)= \mu$. Here $\lambda$ and $\mu$ are arbitrarily preassigned values. By choosing particular $\Omega$, we obtain the precise variability regions of $\log f'(z_0)$ for other well-known subclasses of analytic and univalent functions., Comment: 13 pages

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., Comment: 16

The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ and $\mathcal{S}$ denote the class of functions $f\in\mathcal{A}$ which are univalent ({\it i.e.}, one-to-one). In 1960s, L. Zalcman conjectured that $|a_n^2-a_{2n-1}|\le (n-1)^2$ for $n\ge 2$, which implies the famous Bieberbach conjecture $|a_n|\le n$ for $n\ge 2$. For $f\in \mathcal{S}$, Ma \cite{Ma-1999} proposed a generalized Zalcman conjecture $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1) $$ for $n\ge 2, m\ge 2$. 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 $\mathcal{F}$ denote the class of functions $f\in \mathcal{A}$ satisfying ${\rm Re\,}(1-z)^{2}f'(z)>0$ in $\mathbb{D}$. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{U}$ using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{F}$ for the initial coefficients., Comment: 15 pages