Harmonic Function: Analyticity, Compactness and Minimum Value
Response to the following problem:
Let ƒ(z)=u(x,y)+iv(x,y) be a function that is analytic and not constant throughtout a bounded domain D and continuous on its boundary ∂D (here domain is an open connected set).
Prove, by considering g(z)=eƒ(z) , that the component function u(x,y) has a minimum value in the compact region DU∂D which occurs on ∂D and never in D.
Use this result to formulate and prove a minimum principle for harmonic functions.