Inequalities for harmonic polynomials in two and three dimensions

Abstract

In what follows we discuss certain inequalities involving harmonic polynomials of two and of three variables, that is, polynomials u(x, y) and u(x, y, z) which satisfy Laplace's equation uXX+u,,=O and uxx+uy+uZZ=0, respectively. The inequalities in question furnish bounds for these polynomials and for their derivatives under proper conditions. Of particular interest are inequalities of the type of S. Bernstein's theorem, as discussed by the second author in a recent paper [7 ] ('). The first part of the present paper deals with the two-dimensional and the second part with the three-dimensional case. Of fundamental importance throughout the paper is an interpolation formula which is stated and proved in ?2 of Part I. In fact the results of Part I may be regarded as systematic applications of this formula. Several inequalities of this part are generalizations and refinements of earlier theorems. Most of the problems of the second part are new. In an Appendix we consider a generalization of the main problem treated in Part II, and another problem which deals with ellipses and is only in loose relationship with the other topics considered in the present paper.