The Maximum Modulus Principle

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 $f \in H\left(\mathrm{II}^{+}\right)$, where $\mathrm{II}^{+}$is the upper half plane, and $\left|f\right| \leq 1$. How large can $\left|f^{\prime }(i)\right|$ be? Find the extremal functions. (Compare the discussion in Sec. 12.5).

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

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 $f \in H(\Omega )$, $\Gamma$ is a cycle in $\Omega$ such that $\operatorname {Ind}_{\Gamma }(\alpha ) = 0$ for all $\alpha \notin \Omega ,\left|f(\zeta )\right| \leq 1$ for every $\zeta \in \Gamma^{*}$, and $\operatorname {Ind}_{\Gamma }(z) \neq 0$. Prove that $\left|f(z)\right| \leq 1$.

In the proof of Theorem 12.8 it was tacitly assumed that $M(a) > 0$ and $M(b) > 0$. Show that the theorem is true if $M(a) = 0$, and that then $f(z) = 0$ for all $z \in \Omega$.

If $0 < R_{1} < R_{2} < \infty$, let $A\left(R_{1}, R_{2}\right)$ denote the annulus

There is a vertical strip which the exponential function maps onto $A\left(R_{1}, R_{2}\right)$. Use this to prove Hadamard's three-circle theorem: If $f \in H\left(A\left(R_{1}, R_{2}\right)\right)$, if

and if $R_{1} < a < r < b < R_{2}$, then

For which $f$ does equality hold in this inequality?

Let $\Pi$ be the open right half plane $(z \in \Pi$ if and only if $\mathrm{Re}\left(z\right) > 0)$. Suppose $f$ is continuous on the closure of $\Pi (\mathrm{Re}\left(z\right) \geq 0), f \in H(\Pi )$, and there are constants $A < \infty$ and $\alpha < 1$ such that

for all $z \in \Pi$. Furthermore, $\left|f(i y)\right| \leq 1$ for all real $y$. Prove that $\left|f(z)\right| \leq 1$ in $\Pi$. Show that the conclusion is false for $\alpha = 1$. How does the result have to be modified if $\Pi$ is replaced by a region bounded by two rays through the origin, at an angle not equal to $\pi$?

Let $\Pi$ be the open right half plane. Suppose that $f \in H(\Pi )$, that $\left|f(z)\right| < 1$ for all $z \in \Pi$, and that there exists $\alpha ,-\pi / 2 < \alpha < \pi / 2$, such that

Prove that $f = 0$. Hint: Put $g_{n}(z) = f(z) e^{n z}, n = 1,2,3, \ldots$. Apply Exercise 9 to the two angular regions defined by $-\pi / 2 < \theta < \alpha , \alpha < \theta < \pi / 2$. Conclude that each $g_{n}$ is bounded in $\Pi$, and hence that $\left|g_{n}\right| < 1$ in $\Pi$, for all $n$.

Suppose $\Gamma$ is the boundary of an unbounded region $\Omega , f \in H(\Omega ), f$ is continuous on $\Omega \cup \Gamma$, and there are constants $B < \infty$ and $M < \infty$ such that $\left|f\right| \leq M$ on $\Gamma$ and $\left|f\right| \leq B$ in $\Omega$. Prove that we then actually have $\left|f\right| \leq M$ in $\Omega$.

Suggestion: Show that it involves no loss of generality to assume that $U \cap \Omega = \varnothing$. Fix $z_{0} \in \Omega$, let $n$ be a large integer, let $V$ be a large disc with center at 0, and apply the maximum modulus theorem to the function $f^{n}(z) / z$ in the component of $V \cap \Omega$ which contains $z_{0}$.

Let $f$ be an entire function. If there is a continuous mapping $\gamma$ of $[0,1)$ into the complex plane such that $\gamma (t) \rightarrow \infty$ and $f(\gamma (t)) \rightarrow \alpha$ as $t \rightarrow 1$, we say that $\alpha$ is an asymptotic value of $f$. Prove that every nonconstant entire function has $\infty$ as an asymptotic value.

Suggestion: Let $E_{n} = \left\{ z:\left|f(z)\right| > n\right\}$. Each component of $E_{n}$ is unbounded (proof?) and contains a component of $E_{n+1}$, by Exercise 11.

Show that exp has exactly two asymptotic values: 0 and $\infty$. How about $\sin$ and $\cos$? Note: $\sin z$ and $\cos z$ are defined, for all complex $z$, by

If $f$ is entire and if $\alpha$ is not in the range of $f$, prove that $\alpha$ is an asymptotic value of $f$ unless $f$ is constant.

Suppose $f \in H(U)$. Prove that there is a sequence $\left\{ z_{n}\right\}$ in $U$ such that $\left|z_{n}\right| \rightarrow 1$ and $\left\{ f\left(z_{n}\right)\right\}$ is bounded.

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 $z \in \Omega$.

Let $\Phi$ be the set of all $f \in H(U)$ such that $0 < \left|f(z)\right| < 1$ for $z \in U$, and let $\Phi_{c}$ be the set of all $f \in \Phi$ that have $f(0) = c$. Define

Find $M$, and $M(c)$ for $0 < c < 1$. Find an $f \in \Phi$ with $f^{\prime }(0) = M$, or prove that there is no such $f$. Suggestion: $\log f$ maps $U$ into the left half plane. Compose $\log f$ with a properly chosen map that takes this half plane to $U$. Apply the Schwarz lemma.