Overview
The research paper, published on arXiv, focuses on extending Bohr-Rogosinski type inequalities, which traditionally apply to analytic functions in the complex plane ($\mathbb{C}$), to the domain of holomorphic mappings. Specifically, it considers mappings whose values reside in higher-dimensional complex spaces.
The study primarily establishes the general Bohr-Rogosinski radius for holomorphic mappings. This is first addressed for mappings with values in the closure of the unit polydisc within $\mathbb{C}^n$. Subsequently, the research explores analogous problems for holomorphic mappings, where their values are contained within the closure of the unit ball of a JB$^*$-triple.
Research Context
The foundation of this work lies in existing Bohr-Rogosinski type inequalities applicable to analytic functions defined in $\mathbb{C}$. These inequalities provide bounds related to the coefficients of such functions and the domains within which certain properties hold.
The current research aims to generalize these established inequalities beyond the one-dimensional complex plane. This extension involves considering functions that are holomorphic, a generalization of analytic functions, and whose output values are elements of multi-dimensional complex spaces or more abstract mathematical structures like JB$^*$-triples.
Approach
The methodology involves two primary extensions:
Extension to $\mathbb{C}^n$
The initial phase of the research focuses on holomorphic mappings that take values in the closure of the unit polydisc within $\mathbb{C}^n$. For these mappings, the study aims to derive and present a generalized Bohr-Rogosinski radius. This radius quantifies a specific property or bound related to the mapping within this multi-dimensional complex space.
Extension to JB$^*$-triples
Following the analysis in $\mathbb{C}^n$, the research further extends its scope to a more abstract algebraic structure known as a JB$^*$-triple. For holomorphic mappings whose values are contained within the closure of the unit ball of such a triple, the corresponding Bohr-Rogosinski type problems are considered and addressed. This involves adapting the concepts and methods developed for $\mathbb{C}^n$ to this more general setting.
A key characteristic mentioned for all radii derived in this study is their optimality.
Findings
- The research successfully extends Bohr-Rogosinski type inequalities from analytic functions in $\mathbb{C}$ to holomorphic mappings that operate in higher-dimensional complex spaces.
- A general Bohr-Rogosinski radius has been obtained for holomorphic mappings whose values lie within the closure of the unit polydisc in $\mathbb{C}^n$.
- Corresponding Bohr-Rogosinski problems have been considered and resolved for holomorphic mappings where their values are situated within the closure of the unit ball of a JB$^*$-triple.
- All radii determined within this study are stated to be optimal.