On the Asymptotic Growth of Entire Monogenic Functions

Abstract

In this paper we analyze the behavior of growth of entire monogenic functions in higher dimensional Euclidean spaces. Generalizations of growth orders, the maximum term and of the central index to Clifford analysis provide the basic tools for our analysis. We obtain generalizations of some Valiron's inequalities for transcendental entire monogenic functions. Further to this an asymptotic relation between the growth of a monogenic function and their iterated radial derivatives is established.

On a Generalization of Valiron’s Inequality for k-hypermonogenic Functions on Upper Half-Space

“…The study of the Taylor coefficients of an entire monogenic solution was the basis for the asymptotic growth analysis of entire solutions to partial differential equations (PDEs) related to the generalized Euclidean Dirac and CauchyRiemann operator, see for example [3,4,5,8]. In the context of hypermonogenic functions defined on the upper half-space the study of the Taylor coefficients is replaced by studying their Fourier images, see [6].…”

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

“…For the applications that we are going to develop in this paper it suffices to use the following weaker version which also provides a correction to Theorem 5.2 from [8]: 0n is a left monogenic function in B(0, 1), then for all 0 < r < R < 1:…”

“…As described in [8], in the case of entire monogenic functions the set of Fueter polynomials turned out to be the particular set of entire monogenic functions that have growth order equal to zero. In the case of monogenic Taylor series of finite convergence radius, the class of functions with growth order zero is replaced by the set of hypercomplex meromorphic Mittag-Leffler series with isolated singularities discretely placed on the boundary of the unit ball.…”

“…In the recent papers [5,6,8] the fundamentals of a generalized WimanValiron theory for general entire monogenic functions were established. In particular, generalizations of the maximum term and of the central index to Clifford analysis were introduced for general entire monogenic functions.…”

Some Riemann boundary value problems in Clifford analysis

“…Thus, it is very important to deduce a generalized Liouville theorem in Clifford analysis, too. In [36,37], a generalized Liouville theorem for k-monogenic functions was obtained under some growth conditions. This is a very important result.…”