Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 253690, 6 pages
doi:10.1155/2010/253690
Research Article

Slow Growth for Universal Harmonic Functions

M. Carmen Gómez-Collado,1 Félix Martínez-Giménez,1 Alfredo Peris,1 and Francisco Rodenas2

1IUMPA, Departament de Matemàtica Aplicada, Universitat Politècnica de València, Edifici 7A, 46022 València, Spain
2ISIRM, Departament de Matemàtica Aplicada, Universitat Politècnica de València, ETS Arquitectura, 46022 València, Spain

Received 8 April 2010; Revised 3 June 2010; Accepted 17 June 2010

Academic Editor: Stevo Stevic

Copyright © 2010 M. Carmen Gómez-Collado et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given any continuous increasing function ϕ:[0,+∞[→]0,+∞[ such that lim⁡t→∞log⁡ϕ(t)/log⁡t=+∞, we show that there are harmonic functions H on ℝN satisfying the inequality |H(x)|≤ϕ(∥x∥) for every x∈ℝN, which are universal with respect to translations. This answers positively a problem of D. H. Armitage (2005). The proof combines techniques of Dynamical Systems and Operator Theory, and it does not need any result from Harmonic Analysis.