3.4 Further Properties of Holomorphic Functions
A useful corollary of Theorem 3.1.8 is the Maximum Modulus Principle.
Before the theorem, we first introduce the mean-value property of holomorphic functions.
Lemma 3.4.1.
Let be open and simply connected, and let be holomorphic. Then and such that , is the average of where is uniform. In other words,
Since the real and imaginary parts of holomorphic functions are real-valued harmonic functions, they also satisfy the mean-value property. Furthermore, if a real continuous function satisfies the mean-value property, it is harmonic (to be proved in Theorem 3.6.2). This equivalence allows for the alternative definition of harmonic functions.
Theorem 3.4.1 (Maximum Modulus Principle).
Let be holomorphic on an open connected region . If and an open neighborhood of such that , , then is a constant function on .
Proof.
Assume that exists. We will first prove that the set
is all of . This is equivalent to proving that is nonempty, open, and closed in .
Since , the first condition is satisfied (nonemptiness). For any sequence converging to some , by the continuity of ,
and . Thus, contains all of its accumulation points in and is therefore closed (if , then it is no longer relevant; we are concerned with its relative closedness in ).
Since and are both open, , such that . By Lemma 3.4.1, ,
It follows that all inequalities above are equalities, or that
From the equality of the last two integrals,
Since this integrand is strictly non-negative, we have equality. Thus, , such that . In other words, every has an open neighborhood that also lies in . Therefore, is open and as it is a nonempty clopen subset. Since is nonempty and open, it has an accumulation point in . It follows that over by the Identity Theorem (Theorem 3.3.3).
Remark.
If is holomorphic and non-constant on an open region , then for any compact set , the maximum of in lies on . Otherwise, would attain a maximum at some , and contradict the statement of Theorem 3.4.1 under the assumption of being non-constant.
A similar theorem exists for real-valued harmonic functions. The proof follows in the same way as the one for holomorphic functions. We will state it formally below.
Theorem 3.4.2.
Let be open and connected and let be harmonic. Suppose that and a neighborhood of such that either
Then is constant on .
By nature of the proof, the result holds for any continuous function satisfying the mean-value property.