The Maximum Modulus Principle

Applying Cauchy's integral to the function $\left(f(z)\right)^{n}$, where $n$ is an integer tending to $\infty$, prove the theorem: if $f(z)$ is regular inside a contour $C$, and on the contour the inequality $\left|f(z)\right| \leqslant M$ holds, then it remains valid inside the contour as well.

Prove the theorem: if $f(z)$ is regular inside the disk $\left|z\right| \leqslant R$, and $\left|f(z)\right| < M$, while $f(0) = 0$, then for $\left|z\right| < R$ the inequality $\left|f(z)\right| < \dfrac {M}{R}\cdot \left|z\right|$ holds, and equality at even one point is possible only when $f(z)$ has the form $f(z) = \dfrac {M}{R}e^{i\alpha }z$. (Schwarz's Lemma.)

Prove the theorem: if $f(z)$ is regular for $r < \left|z\right| < R$, then for $r < \rho < R$ the identity

holds, where $a_{n}$ is a coefficient of the Laurent series.

Prove the theorem: if $f(z) = \sum_{n=0}^{\infty } a_{n}z^{n}$ converges for $\left|z\right| < R$, and moreover $\left|f(z)\right| \leqslant M$, then $\sum \left|a_{n}\right|^{2}R^{2n} \leqslant M^{2}$.

Prove that if the identity $w = \sum_{n=0}^{\infty } a_{n}z^{n}$ gives a mapping of the disk $\left|z\right| < R$ onto an area $S$, then $S = \pi \sum_{n=0}^{\infty } n\left|a_{n}\right|^{2}R^{2n}$.

Prove the theorem: if $f(z)$ is regular in the disk $\left|z\right| < R$ and is not constant, then the function

increases together with $r$ for $0 < r < R$.

Prove the theorem: among all functions $f(z)$, regular inside the disk $\left|z\right| < R$ and satisfying the conditions $\int_{0}^{2\pi } \left|f(Re^{\varphi i})\right|^{2}\, d\varphi = M$, the linear function gives the smallest area onto which the disk $\left|z\right| < R$ is mapped.

Prove the theorem: for a function $f(z)$, regular for $r < \left|z\right| < R$, the quantity

is a convex function of $\ln \rho$, i.e., the inequality holds (for $m > 0$, $n > 0$):

Prove the theorem: if $f(z)$, besides the previous conditions (problem 5652), is also nonzero for $r < \left|z\right| < R$, then the quantity

is likewise, for any $\sigma > 0$, a convex function of $\ln \rho$.

Show that the condition $f(z) \neq 0$ in the previous problem (problem 5653) can be dispensed with.

Prove the theorem: if $f(z)$ is regular for $\left|z\right| < R$, then the quantity $I_{\sigma }(\rho )$ (problem 5653) increases together with $\rho$.

Prove the theorem: if $f(z)$ is regular for $r < \left|z\right| < R$, and $M(\rho )$ is the maximum modulus of $f(\rho e^{\varphi i})$, then for $r < \rho < R$ the quantity $M(\rho )$ is a convex function of $\ln \rho$ (Hadamard's Three-Circles Theorem).

Prove the theorem: if a polynomial $f(z)$ of degree $n$ does not exceed $M$ in modulus on the interval $-1 \leqslant z \leqslant 1$, then for any $z$ the inequality $\left|f(z)\right| \leqslant M(a+b)^{n}$ holds, where $a$ and $b$ are the semi-axes of the ellipse passing through the point $z$ and having its foci at the points $z = \pm 1$.

Prove the theorem: if for $\left|z\right| < 1$ the function $f(z)$ is regular and $\left|f(z)\right| < M$, while $f(\alpha ) = 0$, where $\left|\alpha \right| < 1$, then for $\left|z\right| < 1$ the inequality

holds.

Prove the theorem: if $f(z)$ is regular for $\left|z\right| < 1$; $f(0) \neq 0$; $\left|f(z)\right| < M$, and $f(z) = 0$ at $z = z_{1}, z_{2}, z_{3}, \ldots$, then the infinite product $\prod_{\nu =1}^{\infty } \left|z_{\nu }\right|$ converges.

Prove the theorem: if for $\left|z\right| < 1$ the function $f(z)$ is regular and bounded, and for $a < \varphi < b$ the quantity $f(re^{\varphi i})$ tends uniformly to zero as $r \to 1$, then $f(z)$ is identically zero.

Prove the theorem: if the sequence $f_{n}(z)$, for $\left|z\right| \leqslant 1$, is uniformly bounded and tends to $0$ as $n \to \infty$ on a set of points having the origin as a limit point, then $f_{n}(z) \to 0$ as $n \to \infty$ at any interior point of the disk $\left|z\right| < 1$.

Prove the theorem: if the functions $f_{n}(z)$ are regular inside a contour $C$ and uniformly bounded on it, and inside the contour there is a limit point of points at which $f_{n}(z) \to 0$ as $n \to \infty$, then $f_{n}(z) \to 0$ at every point inside the contour as well. (Vitali's Theorem.)

Prove the theorem: if the functions $f_{n}(z)$ are regular in the disk $\left|z\right| < 1$, have no roots there, and are bounded in modulus by $1$, and, moreover, $f_{n}(0) \to 0$ as $n \to \infty$, then throughout the interior of the disk $\left|z\right| < 1$ the functions $f_{n}(z) \to 0$ as $n \to \infty$.

Let $z_{\nu }$ be the roots of an entire function $f(z)$, and

while for the function $f(z)$ the inequality

holds. Prove that for $\left|z\right| \leqslant 3R$

Prove that, under the same conditions (problem 5664), the series $\displaystyle \sum_{\nu =1}^{\infty } \frac{1}{\left|z_{\nu }\right|^{n}}$ converges for $n > \sigma$.

Under the same conditions (problem 5664), prove the Weierstrass infinite-product expansion:

where $g(z)$ is some entire function, and

Let

satisfy, in the disk $\left|z\right| \leqslant R$, the condition: $\mathrm{Re}\left(f(z)\right) \leqslant M$, where $M > 0$. Then

Considering, under the same conditions (problem 5667), the function

where $\omega = e^{\frac{2\pi i}{n}}$, prove the inequality:

Under the same condition on the growth of $f(z)$ as before (see problem 5664), prove that for $\left|z\right| \leqslant 2R$, and hence also for $\left|z\right| \leqslant R$, we have:

where in the product $\left|z_{\nu }\right| \leqslant R$.

Using the fact that $\ln \psi (z)$ is a regular function of $z$ for $\left|z\right| \leqslant R$ (problem 5669), prove that at $z = 0$ the quantity

is, in absolute value, less than $\dfrac {c}{R^{n-\sigma }}$, where $c$ is a corresponding constant.

Prove Hadamard's theorem: if $f(z)$ satisfies, for every $z$, the condition $\left|f(z)\right| < Ce^{\left|z\right|^{\sigma }}$, where $\sigma < n$ and $f(0) \neq 0$, then the Weierstrass expansion

holds, where $g(z)$ is a polynomial whose degree is less than $n$.

Prove Carathéodory's theorem: if $f(z) = \sum_{n=0}^{\infty } a_{n}(z-a)^{n}$ is regular in the disk $\left|z-a\right| \leqslant R$ and there $\mathrm{Re}\left(f(z)\right) < M$, where $M > 0$, then for $\left|z-a\right| = r < R$ the inequality

holds.

Prove Lindelöf's theorem: if $\left|f(z)\right| \leqslant C$ on the edges of the region $-a < x < a$, and inside this region $\left|f(z)\right| < e^{\lambda \left|y\right|}$, where $\lambda$ is some constant, then also inside the region $\left|f(z)\right| \leqslant C$, and moreover equality at an interior point is possible only when $f(z) \equiv C$.