Editing Potential theory (section)

==Inequalities==
A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the [[maximum principle]]. Another important result is [[Harmonic functions#Liouville's theorem|Liouville's theorem]], which states the only bounded harmonic functions defined on the whole of '''R'''<sup>n</sup> are, in fact, constant functions. In addition to these basic inequalities, one has [[Harnack's inequality]], which states that positive harmonic functions on bounded domains are roughly constant.

One important use of these inequalities is to prove [[Limit of a sequence|convergence]] of families of harmonic functions or sub-harmonic functions, see [[Harnack's theorem]]. These convergence theorems are used to prove the [[Existence theorem|existence]] of harmonic functions with particular properties.<ref name=Garabedian>{{cite journal|last1=Garabedian|first1=P. R.|last2=Schiffer| first2=M.|authorlink1=Paul Garabedian|authorlink2=Menahem Max Schiffer| title=On existence theorems of potential theory and conformal mapping|journal=[[Annals of Mathematics]]|jstor=1969517|doi=10.2307/1969517|year=1950|volume=52|number=1|pages=164–187}}</ref>