On harmonic functions with integrable maximum modulus

On the Maximum Principle for Harmonic Functions

Some generalizations of the maximum principle for harmonic functions are discussed. §

On the Maximum Modulus of Polynomials. Ii

Let f(z) := ∑n ν=0 aνz ν be a polynomial of degree n having no zeros in the open unit disc, and suppose that max|z|=1 |f(z)| = 1. How small can max|z|=ρ |f(z)| be for any ρ ∈ [0 , 1)? This problem was considered and solved by Rivlin [4]. There are reasons to consider the same problem under the additional assumption that f ′(0) = 0. This was initiated by Govil [2] and followed up by the present ...

On the Structure of Maximum Modulus Algebras

i.e., if every fER attains its maximum modulus on C. As a matter of convenience, we shall, in this paper, abbreviate "maximum modulus algebra on K" to "ilf-algebra." Examples of Af-algebras which come to mind immediately are (a) the algebra ft consisting of all functions which are continuous on K and analytic in U, (b) any subalgebra of ft, (c) any algebra (B which is equivalent to an algebra o...

More on Harmonic Functions

Computing with Harmonic Functions

Earlier versions of this software focused on algorithms arising from the material in the book Harmonic Function Theory [ABR] by Sheldon Axler, Paul Bourdon, and Wade Ramey. That book is still the source for many of the algorithms used in the HFT10.m package, but the goal of the package has expanded to include additional symbolic manipulations involving harmonic functions. The HFT10.m package ca...