Step 1

Mean Value Property

We motivate and define the Mean Value Property of a complex-valued function.

Step 2

Maximum Modulus Property

As a next step we define the Maximum Modulus Property.

Step 3

Connection between the Mean Value Property and the Maximum Modulus Property

It is natural now to connect the Mean Value Property and the Maximum Modulus Property

Step 4

Maximum Principle

We now state and prove the important and well-known maximum principle and we use it to prove the important and less-known minimum principle.

Step 5

Schwarz Lemma

We conclude this session with Schwarz lemma which has many applications.