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

Abstract

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.

Cited by in corpus (2)

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