## The Maximum Principle

Let a function be analytic on a bounded region and continuous on Then there exists (the boundary of ) such that for (1) (the interior together with the boundary of ).

This is the Maximum Principle

Proof

Note that ( the interior together with the boundary of R) is closed. R is also bounded and so for some M &gt;0.

Also since all points outside are exterior to R and so Thus is a compact set. By the Extreme Value Theorem, as is continuous on there exists such that for If then the proof is complete. Otherwise so must be constant on by the Local Maximum Principle. It follows by the continuity of on that is constant on also. In fact if is any point of then there exists a sequence in such that and hence by the continuity of at Thus if say, for then for so that also.

Hence is constant on so that (1) holds for any point of If the function is non – constant and analytic on then (1) can be strengthened to for In fact, if for some then must have a local maximum at which is not possible by the Local Maximum Theorem.