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

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

2010-04-07

arxiv.org/abs/1004.1188v1

Mathematics

Complex Variables

Scientific paper

The Bohr theorem states that any function $f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}$, analytic and bounded in the open unit disk, obeys the inequality $\sum_{n=0}^{\infty} |a_{n}| |z|^{n} < 1$ 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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