Bohr Phenomena for Slice Regular Functions over Quaternions Achieves Refined Inequalities for All Values

The behaviour of functions extending from complex numbers to quaternions, a mathematical system extending complex numbers, presents a fascinating challenge for mathematicians, and recent work by Sabir Ahammed, Molla Basir Ahamed, and Ming-Sheng Liu addresses a key aspect of this challenge. The researchers investigate ‘Bohr phenomena’ for a specific class of quaternion functions called ‘slice regular functions’, effectively establishing a mathematical boundary on how these functions can behave. This work extends the well-known Bohr inequality, a fundamental result in complex analysis, to the more complex realm of quaternions, and importantly, provides improved and refined versions of this inequality. These advancements significantly deepen our understanding of quaternion functions and have implications for various fields including signal processing, quantum mechanics, and engineering applications that utilise these extended mathematical systems.

Bohr’s Inequality and Slice Regular Functions

This research investigates Bohr’s inequality and its extensions to various function spaces and algebras, focusing particularly on slice regular functions, which generalize holomorphic functions to quaternions and other non-commutative algebras. Scientists explore Bohr-type inequalities in multiple complex variables, quaternion algebras, and related settings. Key concepts include Bohr’s inequality, a classical result bounding the coefficients of power series representing bounded analytic functions, and slice regular functions, which are regular on slices of quaternion or octonion algebras. The study explores extensions of Bohr’s inequality to functions of multiple complex variables and focuses on its behavior in quaternion and octonion algebras. Researchers present new results refining existing Bohr inequalities or extending them to broader classes of functions, illustrating these results with specific examples. This is a significant research paper contributing to the ongoing development of Bohr’s inequality and its applications to complex analysis and non-commutative algebras.

Bohr Inequalities for Slice Regular Functions

This work establishes refined Bohr inequalities for slice regular functions, extending a classical result from complex analysis to the quaternionic setting. Researchers first investigated Bohr-type inequalities for slice starlike and slice close-to-convex functions, laying the groundwork for more general results. The study generalized the Bohr inequality to apply to slice regular functions defined on the open unit ball, demonstrating its validity in this extended mathematical space. To achieve these results, scientists carefully analyzed the properties of slice regular functions, which generalize holomorphic functions to quaternions.

The approach involved establishing bounds on the growth of these functions and their derivatives, leveraging the unique algebraic structure of quaternions. Researchers developed novel techniques to control the behavior of slice regular functions, enabling them to derive improved versions of the Bohr inequality. This inequality provides a precise constraint on the coefficients of the function’s power series representation, offering a deeper understanding of their interrelationship.

Refined Bohr Inequality for Quaternion Functions

Scientists have established a refined Bohr inequality for slice regular functions, extending this important mathematical concept beyond complex numbers to the realm of quaternions. This work builds upon the historical development of Bohr’s theorem, initially focused on Dirichlet series and analytic functions, and generalizes it to functions with quaternion variables. The team rigorously demonstrated that the inequality holds for slice starlike and slice close-to-convex functions over quaternions, establishing a foundational result for this expanded mathematical framework. The research confirms that the refined Bohr inequality applies to slice regular functions defined on the open unit ball of quaternions, providing a precise mathematical boundary for their behavior. Specifically, the research confirms that for all functions within this defined space, the inequality holds true, demonstrating a consistent and predictable pattern. Further investigations focused on functions where the real part of the function’s value is less than or equal to one for all points within the defined space, delivering a precise mathematical statement about their behavior.

Sharp Bohr Inequalities for Slice Regular Functions

This research significantly advances the understanding of slice regular functions, which represent a generalization of holomorphic functions to quaternions and Clifford algebras. The team established new forms of the Bohr inequality for slice starlike and slice close-to-convex functions, extending these important results to quaternionic settings. Furthermore, they derived improved and refined versions of the Bohr inequality specifically for slice regular functions defined on open unit balls. These inequalities provide bounds on the growth of functions and are crucial for understanding their behavior.

The work demonstrates the sharpness of these inequalities, confirming that the established bounds are optimal and cannot be further improved. This was achieved through careful analysis and the construction of specific slice regular functions that meet the bounds. The researchers also investigated conditions under which these inequalities hold, providing a more complete picture of their applicability.