The Maximum Principle

Let the function $f(z)$ be regular in a domain $G$. Prove that if, for every sequence of points $z_{n} \in G$ converging to some boundary point of the domain $G$, the inequality

holds, then either the inequality $\left|f(z)\right| < M$ holds at every interior point of the domain $G$, or $f(z) \equiv M e^{i \varphi }$.

Let the function $f(z)$ be regular in the disk $\left|z\right| < 1$ and satisfy in this disk the inequality $\left|f(z)\right| < M$. Prove that if $f(0) = 0$, then the function $f(z)$ also satisfies in the disk $\left|z\right| < 1$ the stronger inequality

and if, at even one point $z_{0}, 0 < \left|z_{0}\right| < 1$, the equality $\left|f\left(z_{0}\right)\right| = M\left|z_{0}\right|$ holds, then

where $\varphi$ is a real constant. (Schwarz's lemma.)

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$ and satisfy there the inequality $\left|f(z)\right| < M$, with $f(0) = 0$. Prove that

and equality is possible only for the function $f(z) = M e^{i \varphi } \cdot \frac{z}{R}$.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$, satisfy there the inequality $\left|f(z)\right| < M$, and vanish at some point $z_{0}$ of this disk. Prove the inequalities

Let the function $f(z)$ be regular in the strip $\left|\mathrm{Re}\left(z\right)\right| < \pi / 4$, satisfy there the inequality $\left|f(z)\right| < 1$, and vanish at the point $z = 0$. Prove that $\left|f(z)\right| \leqslant \left|\operatorname {tg} z\right|$ in this strip.

Let the function $f(z)$ be regular for $\mathrm{Re}\left(z\right) > 0$, satisfy there the inequality $\left|f(z)\right| < 1$, and vanish at the points $z_{1}, z_{2}, \ldots , z_{m}$. Prove that

Let the function $f(z)$ be regular and bounded in the half-plane $\mathrm{Re}\left(z\right) > 0$, and vanish at a sequence of points $\left\{ z_{n}\right\} , z_{n} \rightarrow \infty$, of this half-plane. Prove that either $f(z) \equiv 0$, or the series $\sum_{n = 1}^{\infty } \mathrm{Re}\left(\frac{1}{z_{n}}\right)$ converges.

Let the function $f(z)$ be regular and bounded in the disk $\left|z\right| < R$, and vanish at a sequence $\left\{ z_{n}\right\}$ of points of this disk. Prove that either $f(z) \equiv 0$, or the series $\sum_{n = 1}^{\infty }\left(R-\left|z_{n}\right|\right)$ converges.

Let the function $f(z)$ be regular in the disk $\left|z\right| < 1$, satisfy there the inequality $\left|f(z)\right| < M$, with $f(0) = w_{0}$. Prove the inequality

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$ and satisfy the inequalities $\left|f(z)\right| < M,\left|f(0)\right| \leqslant m < M$. Prove the inequality

Let $P(z)$ be a polynomial of degree $n$, and let $M(r) = \max_{\left|z\right| = r}\left|P(z)\right|$. Prove that for $0 < r_{1} < r_{2}$ the inequality

holds, and equality, for even one pair of values $r_{1}$ and $r_{2}$, is possible only for a polynomial of the form $P(z) = a z^{n}$.

Let $P(z) = z^{n}+a_{1} z^{n-1}+\ldots +a_{n}$. Prove that either the inequality $\left|P(z)\right| > 1$ holds at some point of the circle $\left|z\right| = 1$, or $P(z) \equiv z^{n}$.

Let $P(z)$ be a polynomial of degree $n$ satisfying, on the interval $(-1,1)$, the inequality $\left|P(z)\right| \leqslant M$. Prove that at every point $z_{0}$ lying outside this interval the inequality $\left|P\left(z_{0}\right)\right| \leqslant M(a+b)^{n}$ holds, where $a$ and $b$ are the semi-axes of the ellipse with foci $-1$ and $1$ passing through the point $z_{0}$.

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 $G$.

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 $G$.

Let the functions $f_{1}(z), \ldots , f_{m}(z)$ be regular in a domain $G$, and let $M = \varlimsup_{z \rightarrow \partial G}\left\{ \left|f_{1}(z)\right|+\ldots +\left|f_{m}(z)\right|\right\}$. Prove that if at least one of the $f_{k}(z)$ is not identically constant, then at every point of $G$ the inequality $\left|f_{1}(z)\right|+\ldots +\left|f_{m}(z)\right| < M$ holds.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$, and let $m$ be a positive integer. Prove that if the function $f(z)$ is not identically constant, then the function

monotonically increases for $0 \leqslant r < R$.

Prove that a function $u(z)$, harmonic in a domain $G$, is subharmonic in this domain.

Let the function $u(z)$ be harmonic in a domain $G$. Denote $M = \sup \left\{ u(z): z \in G\right\} , m = \inf \left\{ u(z): z \in G\right\}$. Prove that if $m < M$, then at every point of the domain $G$ the inequalities $m < u(z) < M$ hold. (The maximum and minimum principle for harmonic functions.)

Prove that the function $u(z) = \left|f(z)\right|$ is subharmonic in a domain $G$, if the function $f(z)$ is regular in the domain $G$.

Let the function $u(z)$ have, in a domain $G \subset \mathbb {C}$, continuous partial derivatives of the second order (with respect to $x = \mathrm{Re}\left(z\right)$ and $y = \mathrm{Im}\left(z\right)$), and let $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} \geqslant 0$ in the domain $G$. Prove that the function $u(z)$ is subharmonic in the domain $G$.

Let the function $f(z)$ be regular in a domain $G$. Prove that the functions $u(z) = \ln \left|f(z)\right|$ and $u(z) = \left|f(z)\right|^{\alpha }, \alpha > 0$, are subharmonic in the domain $G$.

Let the functions $u_{1}(z)$ and $u_{2}(z)$ be subharmonic in a domain $G$. Prove that:

Let the functions $f_{1}(z), \ldots , f_{m}(z)$ be regular in a domain $G$. Prove that for $\alpha > 0$ the function $u(z) = \left|f_{1}(z)\right|^{\alpha }+\ldots +\left|f_{m}(z)\right|^{\alpha }$ is subharmonic in the domain $G$.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$. Prove that the function

is, for any $\alpha > 0$, a monotonically increasing function of $r$ on the interval $0 \leqslant r < R$ (if $f(z) \not\equiv$ const).

Prove that the function

is subharmonic in the strip $a < \mathrm{Re}\left(z\right) < b$ if and only if the function $\varphi (x)$ is convex downward on the interval $a < x < b$.

Prove that the function $u\left(r e^{i \theta }\right) = \varphi (r)$ is subharmonic in the annulus $\rho < \left|z\right| < R$ if and only if the function $\varphi (r)$ is logarithmically convex on the interval $(\rho , R)$, i.e., if for any three values $\rho < r_{1} < r_{2} < r_{3} < R$ the inequality holds

Prove that the function $u\left(r e^{i \theta }\right) = r^{\rho } \varphi (\theta )$ is subharmonic in the angle $\alpha < \theta < \beta$ if and only if the function $\varphi (\theta )$ is trigonometrically $\rho$-convex on the interval $(\alpha , \beta )$, i.e., if for any triple of values $\theta_{1}, \theta_{2}, \theta_{3}$ satisfying the conditions $\alpha < \theta_{1} < \theta_{2} < \theta_{3} < \beta$ and $\theta_{3}-\theta_{1} < \pi / \rho$, the inequality holds

Let the function $f(z)$ be regular in the annulus $\rho < \left|z\right| < R$. Denote

Prove that for $\rho < r_{1} < r_{2} < r_{3} < R$ the inequality holds

(Hadamard's three-circle theorem.)

Let the function $f(z)$ be regular in the annulus $\rho < \left|z\right| < R$. Denote

Prove that for $\rho < r_{1} < r_{2} < r_{3} < R$ the inequality holds

Let the function $f(z)$ be regular in the half-plane $\mathrm{Re}\left(z\right) > 0$ and satisfy the inequality

Prove that the function

is a downward convex function of $\varphi$ on the interval $-\pi / 2 < \varphi < \pi / 2$.