In this paper, we establish several higher-dimensional generalizations of refined Bohr-type inequalities for bounded holomorphic functions mapping into the unit polydisk $\mathbb{P}\Delta(0;1_n)$ in $\mathbb{C}^n$. First, we formulate multidimensional analogues of sharp Bohr-type inequalities originally established by Liu \emph{et al.} [{\it Bull. Sci. Math.} {\bf 173} (2021) 103054], incorporating both squared coefficient terms and area functional components. Second, we provide improved inequalities for a recent multidimensional extension by Ahamed \emph{et al.} [{\it Complex Anal. Oper. Theory} {\bf 20}(6) (2026), 142] by introducing an analogous term corresponding to the area functional. Finally, we extend a refined Bohr-type inequality involving the term $\vert{}f(z)-a_0\vert{}$ to the setting of several complex variables. All the results are shown to be sharp.
In this article, we investigate refined and generalized versions of the Bohr and Bohr--Rogosinski inequalities for a normalized subclass $\mathcal{P}_{\mathcal{H}}^{0}(\alpha)$ ($0 \le \alpha<1$) of univalent close-to-convex harmonic mappings $f = h + \overline{g}$ defined on the open unit disk $\mathbb{D} \subset \mat...
Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fr\'echet derivatives to...
We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enla...
In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally...
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u...
Akram Khan, Sagar Gautam, Manil T. Mohan· 0 citations
Though the sharp $L^{2}$-Caffarelli--Kohn--Nirenberg (CKN) inequalities have been extensively studied in the entire Euclidean spaces, the corresponding problem on domains whose boundary contains the origin remains largely unexplored. We investigate the sharp $L^{2}$-CKN inequalities on half-spaces and orthants $\mathbb...
L. Nguyen, Yukta Lodha, Guo-Zhen Lu et al.· 4 citations
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