arXiv subjects
Publications and source records attributed to Raju Biswas.
The primary objective of this paper is to establish several sharp versions of improved Bohr inequalities, refined Bohr inequalities, and Bohr-Rogosinski inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D}:=\{z\in\mathbb{C}: |z|<1\}$, where $f$ and $g$ are analytic functions in $\mathbb{D}$.
The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In order to achieve these objectives, we employ the non-negative quantity $S_ρ(h)$ and the concept of replacing the initial coefficients of the majorant series by the absolute values of the analytic function and its derivative, as well as other various settings. Moreover, we obtain the sharp Bohr-Rogosinski radius for harmonic mappings in the unit disk by replacing the bounding condition on the analytic function $h$ with the half-plane condition.
The classical Bohr theorem and its subsequent generalizations have become active areas of research, with investigations conducted in numerous function spaces. Let $\{ψ_n(r)\}_{n=0}^\infty$ be a sequence of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^\infty ψ_n(r)$ converges locally uniformly on the interval $[0, 1)$. The main objective of this paper is to establish several sharp versions of generalized Bohr inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\D := \{z \in \mathbb{C} : |z| < 1\}$. To achieve these, we employ the sequence of functions $\{ψ_n(r)\}_{n=0}^\infty$ in the majorant series rather than the conventional dependence on the basis sequence $\{r^n\}_{n=0}^\infty$. As applications, we derive a number of previously published results as well as a number of sharply improved and refined Bohr inequalities for harmonic mappings in $\D$. Moreover, we obtain a convolution counterpart of the Bohr theorem for harmonic mapping within the context of the Gaussian hypergeometric function
Let $\mathcal{B}(\mathcal{H})$ denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces $\mathcal{H}$. In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class $\mathcal{B}(\mathbb{D}, \mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}$ to $\mathcal{B}(\mathcal{H})$ with $\sup_{|z|<1}\Vert f(z)\leq 1$ by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain $Ω\subset\mathbb{C}^n$. All of the results are sharp.
In this paper, we derive the sharp Bohr type inequality for the Cesáro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of bounded analytic functions defined on shifted disks \beas Ω_γ=\left\{z\in\mathbb{C}:\left|z+\fracγ{1-γ}\right|<\frac{1}{1-γ}\right\}\quad\text{for}\quadγ\in[0,1).\eeas
In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Cesáro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Cesáro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.