On Unique Minimal $$\boldsymbol{L}^{\boldsymbol{p}}$$-Norm Harmonic or Holomorphic Function Which Takes Given Value in a Fixed Point

Abstract

First, it will be shown that Banach spaces $$V$$ of harmonic or holomorphic functions with $$L^{p}$$ norm satisfy minimal norm property, i.e., in any set $$V_{z,c}:=\{f\in V\>|\>f(z)=c\},$$ if nonempty, there is exactly one element with minimal norm. Later, it will be proved that this element depends continuously on a deformation of a norm and on an increasing sequence of domains in a precisely defined sense.