Maximum modulus principle (theorem statement)

Let $f(z)$ be a function that is analytic in the unit disc $\mathbb {D} = \left\{ \left|z\right|<1\right\}$. Suppose that $\left|f(z)\right| \leq 1$ in $\mathbb {D}$. Prove that if $f(z)$ has at least two fixed points $z_{1}$ and $z_{2}$, then $f(z) = z$ for all $z \in \mathbb {D}$.

Suppose $h(z)$ is analytic in the whole plane, $h(0) = 3+4i$, and $\left|h(z)\right| \leq 5$ if $\left|z\right|<1$. What is $h'(0)$?

Let $f$ be analytic in the closed unit disc, with $f(-\log 2) = 0$ and $\left|f(z)\right| \leq \left|e^{z}\right|$ for all $z$ with $\left|z\right| = 1$. How large can $\left|f(\log 2)\right|$ be?

Let $f(z)$ be analytic on the open unit disc $\mathbb {D} = \left\{ z \mid \left|z\right|<1\right\}$. Prove that there is a sequence $(z_{n})$ in $\mathbb {D}$ such that $\left|z_{n}\right| \to 1$ and $(f(z_{n}))$ is bounded.

Let $f$ be an analytic function in the unit disc, $\left|z\right|<1$.

Let the function $f$ be analytic in the entire complex plane, and suppose that $f(z)/z \to 0$ as $\left|z\right| \to \infty$. Prove that $f$ is constant.

Let the rational function $f$ in the complex plane have no poles for $\mathrm{Im}\left(z\right) \geq 0$. Prove that

Let $p$ and $q$ be nonconstant complex polynomials of the same degree whose zeros lie in the open disk $\left|z\right|<1$. Prove that if $\left|p(z)\right| = \left|q(z)\right|$ for $\left|z\right| = 1$ then $q = \lambda p$ for a unimodular constant $\lambda$.

Let $f$ be an entire function. Define $\Omega = \mathbb {C} \setminus (-\infty , 0]$, the complex plane with the ray $(-\infty , 0]$ removed. Suppose that for all $z \in \Omega$, $\left|f(z)\right| \leq \left|\log z\right|$, where $\log z$ is the principal branch of the logarithm. What can one conclude about the function $f$?

Let the continuous real-valued function $\varphi$ on the complex plane satisfy the Sub--Mean--Value Property: for any point $z_{0}$,

Prove that $\varphi$ obeys the Maximum Modulus Principle: the maximum of $\varphi$ over any compact set $K$ is attained on the boundary of $K$.

Let $a>0$. Show that the complex function

satisfies $\left|f(z)\right|<1$ for all $z$ in the open left half-plane $\mathrm{Re}\left(z\right)<0$.

Let $f$ be analytic in an open set containing the closed unit disc. Suppose that $\left|f(z)\right| > m$ for $\left|z\right| = 1$ and $\left|f(0)\right| < m$. Prove that $f(z)$ has at least one zero in the open unit disc $\left|z\right| < 1$.

Let $f$ be continuous on the closure $\overline{S}$ of the unit square

and let $f$ be analytic on $S$. If $\mathrm{Re}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ y = 0\right\} \cup \left\{ y = 1\right\} )$, and if $\mathrm{Im}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ x = 0\right\} \cup \left\{ x = 1\right\} )$, prove that $f \equiv 0$ everywhere on $\overline{S}$.

Determine all continuous functions on $\left\{ z \in \mathbb {C} : 0 \leq \left|z\right| \leq 1\right\}$ which are harmonic on $\left\{ z : 0 < \left|z\right| < 1\right\}$ and which are identically $0$ on $\left\{ z \in \mathbb {C} : \left|z\right| = 1\right\}$.

Let $a \in \mathbb {C}$, $\left|a\right| \leq 1$, and consider the polynomial

Show that $\left|P(z)\right| \leq 1$ whenever $\left|z\right| \leq 1$.

Let $f$ be holomorphic in the unit disk $\left\{ \left|z\right| < 1\right\}$, continuous in $\left\{ \left|z\right| \leq 1\right\}$ and $\left|f(z)\right| = 1$ whenever $\left|z\right| = 1$. Prove that $f$ is a rational function.

Let $f$ be a continuous function on $U = \left\{ z : \left|z\right| \leq 1\right\}$ such that $f$ is analytic in $U^\circ$. If $f \equiv 1$ on the half-circle $\gamma = \left\{ e^{i\theta } : 0 \leq \theta \leq \pi \right\}$, prove that $f \equiv 1$ everywhere in $U$.

Let $S$ denote the sector in the complex plane given by $S = \left\{ z : -\frac\pi 4 < \arg z < \frac\pi 4\right\}$. Let $\closure S$ denote the closure of $S$. Let $f$ be a continuous complex function on $\closure S$ which is holomorphic in $S$. Suppose further

1. $\left|f(z)\right| \leq 1$ for all $z$ in the boundary of $S$;

2. $\left|f(x+iy)\right| \leq e^{\varepsilon x}$ for all $x+iy \in S$. Prove that $\left|f(z)\right| \leq 1$ for all $z \in S$.

Let $K$ be a compact, connected subset of $\mathbb {C}$ containing more than one point and let $f$ be a one-to-one conformal map $\mathbb {C} \setminus K$ onto $\Delta = \left\{ z : \left|z\right| < 1\right\}$ with $f(\infty ) = 0$. If $p$ is a polynomial of degree $n$ for which $\left|p(z)\right| \leq 1$ for $z \in K$, prove that

Suppose $f$ and $g$ (non-constant functions) are analytic in a region $G$ and continuous on the closure $\closure G$ of the region. Assume that $\closure G$ is compact. Prove that $\left|f\right| + \left|g\right|$ achieves its maximum value on the boundary of $G$.

Suppose $f(z)$ is an entire function with

Show that $f(z)$ is identically $0$.

Suppose $f$ is analytic on $\left\{ z; 0 < \left|z\right| < 1\right\}$ and

Show that $f \equiv 0$.

Let $f$ be an analytic function on $D = \left\{ z : \left|z\right| < 1\right\}$, $f(D) \subset D$ and $f(0) = 0$.

Let $f(z)$ be a holomorphic function on $\left\{ \left|z\right| < 1\right\}$ and continuous up to $\left\{ \left|z\right| \leq 1\right\}$. Let $M$ be the supremum of $\left|f\right|$ on $\left\{ \left|z\right| \leq 1\right\}$. Let $L$ be the intersection of $\left\{ \left|z\right| \leq 1\right\}$ and $\left\{ \re z = 0\right\}$. Let $m$ be the supremum of $\left|f\right|$ on $L$. Show that

Let $f$ be a bounded analytic function in the right half-plane. Prove that if $f(n) = 0$ for $n = 1,2,3,\ldots$, then $f \equiv 0$.

State whether the following statement is True or False, and prove your assertion.

For each positive integer $n$ there exists an entire function $f_n$ such that

The function $w = f(z)$ is regular in a simply connected domain bounded by the curve $\left|w(z)\right| = c$. Show that $\arg w$ varies monotonically as $z$ traverses the curve $\left|w(z)\right| = c$ in the positive direction.

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 that the linear space is normed with respect to the indicated norm. $H(K)$ — the set of functions analytic inside the open unit disc $K = \left\{ z \in C \mid \left|z\right| < 1\right\}$ that are continuous on its closure $\bar{K}, \left\| x\right\| = \max_{\left|z\right| = 1}\left|z(t)\right|$.

Let $f(z)$ be a regular and single-valued function in the disk $\left|z-z_{0}\right|\leq r$ and $M$ be the maximum of $\left|f(z)\right|$ for $z$ on the circle $\left|z-z_{0}\right|=r$. Then

Equality holds if and only if $f(z)$ is a constant.

The function $f(z)$ is assumed to be regular and single-valued in the domain $\mathfrak {D}$. The maximum of $\left|f(z)\right|$ on the boundary of $\mathfrak {D}$ is denoted by $M$. Then

in the interior of $\mathfrak {D}$ unless $f(z)$ is a constant. (The Maximum Modulus Principle, or briefly Maximum Principle; cf.Chapter 6.)

What does the maximum principle say about the modular graph?

The $n$ points $P_{1},P_{2},\ldots ,P_{n}$ are given in a plane, $P$ is a variable point in this plane. In any domain $\mathfrak {D}$ the function of the point $P$

($\overline{PP_{\nu }}$ is the distance between the points $P$ and $P_{\nu }$) assumes its maximum on the boundary.

Let $f(z)$ be regular, single-valued and non-vanishing in the domain $\mathfrak {D}$. If $f(z)$ is not a constant, $\left|f(z)\right|$ can assume its minimum only at boundary points of $\mathfrak {D}$.

Consider the domain bounded by a closed level line ($\left|f(z)\right|$ is constant along this curve) without selfintersections and lying inside the region where $f(z)$ is regular. It contains at least one zero of $f(z)$ unless $f(z)$ is a constant.

Given $n$ points $P_{1},P_{2},\ldots ,P_{n}$ in a plane in which $P$ varies. The locus of the points $P$ for which the product of the distances

is a ``lemniscate with $n$ foci''. (The standard lemniscate represents the special case $n=2$, cf.problem 4.) Show that a lemniscate with $n$ foci can never consist of more than $n$ separate closed branches.

Let the function $f(z)$ be regular on the disk $\left|z\right|\leq r$ and $z_{0}$, $\left|z_{0}\right|=r$, be a point at which $\left|f(z)\right|$ assumes its maximum. Then $z_{0}\dfrac {f’(z_{0})}{f(z_{0})}$ is real and positive. [Problems 103, 132.]

The maximum of $\left|f(z)\right|$ on the disk $\left|z\right|\leq r$ is $M(r)$.

The maximum $M(r)$ is monotone increasing with $r$ unless $f(z)$ is a constant.

We assume that the function $f(z)$ is regular in the simply connected region $\left|z\right|>R$. $M(r)$ denotes the maximum of $\left|f(z)\right|$ on the circle $\left|z\right|=r$, $r>R$. Then $M(r)$ is also the maximum of $\left|f(z)\right|$ for $\left|z\right|\geq r$ and $M(r)$ is monotone decreasing unless $f(z)$ is a constant.

Let $f(z)$ denote a polynomial of degree $n$; then

Equality is attained only if the polynomial is of the form $cz^{n}$.

Suppose that $f(z)$ is a polynomial of degree $n$ and that

on the real interval $-1\leq z\leq 1$. Then we have

for any $z$ outside this interval; $a$ and $b$ are the semi-axes of the ellipse through $z$ and with foci $-1$ and $1$. What does the proposition imply for $z\to \infty$?

We assume that $f(z)$ is a polynomial of degree $n$, that $E_{1}$ and $E_{2}$ denote two homofocal ellipses with semi-axes $a_{1},b_{1}$ and $a_{2},b_{2}$, $a_{1}<a_{2}$, $b_{1}<b_{2}$. The maximum of $\left|f(z)\right|$ on $E_{1}$ and $E_{2}$ is denoted by $M_{1}$ and $M_{2}$ resp.; then

Derive problems 269 and 270 from this proposition.

Let $L$ be a simple closed continuous curve containing the origin, $\bar{r}$ the outer radius of $L$ and $r_{0}$ the inner radius of $L$ associated with the origin. Let $P(z)$ be a polynomial whose lowest order term is $a_{k}z^{k}$ and whose highest order term is $a_{n}z^{n}$, $P(z) = a_{k}z^{k}+a_{k+1}z^{k+1}+\cdots +a_{n}z^{n}$, and let $M$ denote the maximum of $\left|P(z)\right|$, as $z$ describes the curve $L$. Then

The equality sign holds only if $L$ is a circle with center at the origin and the polynomial in question is a power of $z$ multiplied by a constant.

For every entire function (that is not a constant) the number $\infty$ is an asymptotic value.

Suppose $\Delta$ is a closed equilateral triangle in the plane, with vertices $a, b, c$. Find $\max (\left|z-a\right|\left|z-b\right|\left|z-c\right|)$ as $z$ ranges over $\Delta$.

Suppose $\Omega$ is a bounded region, $\left\{ f_{m}\right\}$ is a sequence of continuous functions on $\bar{\Omega }$ which are holomorphic in $\Omega$, and $\left\{ f_{n}\right\}$ converges uniformly on the boundary of $\Omega$. Prove that $\left\{ f_{n}\right\}$ converges uniformly on $\bar{\Omega }$.

Suppose $\Omega$ is a bounded region, $f \in H(\Omega )$, and

for every sequence $\left\{ z_{n}\right\}$ in $\Omega$ which converges to a boundary point of $\Omega$. Prove that $\left|f(z)\right| \leq M$ for all $z \in \Omega$.