Minimum modulus principle and forced zeros from boundary behavior

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Problem 2.10.22 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.10 "Elementary Properties of Holomorphic Functions"

Suppose $f \in H(\Omega ), \Omega$ contains the closed unit disc, $\left|f(z)\right| > 2$ if $\left|z\right| = 1$ , and $f(0) = 1$ . Must $f$ have a zero in the unit disc?

Problem 2.12.3 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.12 "The Maximum Modulus Principle"

Suppose $f \in H(\Omega )$ . Under what conditions can $\left|f\right|$ have a local minimum in $\Omega$ ?

Problem 2.12.4 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.12 "The Maximum Modulus Principle"

(a)(b)

(a)

Suppose $\Omega$ is a region, $D$ is a disc, $\bar{D} \subset \Omega , f \in H(\Omega ), f$ is not constant, and $\left|f\right|$ is constant on the boundary of $D$ . Prove that $f$ has at least one zero in $D$ .

(b)

Find all entire functions $f$ such that $\left|f(z)\right| = 1$ whenever $\left|z\right| = 1$ .

Problem 10.14 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the function $f(z)$ be regular in a domain $G$ , and let $\inf_{z \in G}\left|f(z)\right| = \mu > 0$ . Prove that either $f(z) \equiv \mu e^{i \varphi }$ , or $\left|f(z)\right| > \mu$ at every interior point of the domain $G$ .

Problem 10.15 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the function $f(z)$ be regular in a domain $G$ and continuous in its closure $\bar{G}$ , with its modulus retaining a constant value on the boundary of the domain $G$ . Prove that if the function $f(z)$ is not identically constant, then it vanishes at some point of the domain $G$ .

References

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.