On the development of Bohr's phenomenon in the context of Quaternionic analysis and related problems

2010/04/07 by J. Morais, Morais, J., K. Guerlebeck +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods #math.CV #msc:30G35

paper · pdf · doi:10.48550/arxiv.1004.1188

arxiv created 2010/04/07 · arxiv updated 2010/04/09

Abstract

The Bohr theorem states that any function f(z) = ∑_n=0^∞ a_n zⁿ, analytic and bounded in the open unit disk, obeys the inequality ∑_n=0^∞ |a_n| |z|ⁿ < 1 in the open disk of radius 1/3, the so-called Bohr radius. Moreover, the value 1/ cannot be improved. In this paper we review some results related to this theorem for the three-dimensional Euclidean space in the setting of quaternionic analysis. The existing results for the Bohr radius will be improved and also some estimates for the hypercomplex derivative of a monogenic function by the norm of the function will be proved.

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• On a three dimensional analogue to the holomorphic z-powers
• Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3

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