Functions on the Unit Disc

Let $\alpha$ be a number in $(0, \pi /2)$. Prove that the function $f(z) = e^{-1/z}$ is uniformly continuous in $S_{\alpha } = \left\{ z \mid 0<\left|z\right| \leq 1, \left|\operatorname {Arg} z\right| \leq \alpha \right\}$, a sector of the complex plane.

Let $a$ and $b$ be nonzero complex numbers and $f(z) = az+bz^{-1}$. Determine the image under $f$ of the unit circle $\left\{ z \mid \left|z\right| = 1\right\}$.

Let $f$ be analytic on and inside the unit circle $C = \left\{ z \mid \left|z\right| = 1\right\}$. Let $L$ be the length of the image of $C$ under $f$. Show that $L \geq 2\pi \left|f'(0)\right|$.

Let

be analytic in the disc $\mathbb {D} = \left\{ z \in \mathbb {C} \mid \left|z\right|<1\right\}$. Assume $f$ maps $\mathbb {D}$ one-to-one onto a domain $G$ having area $A$. Prove

Compute the area of the image of the unit disc $\left\{ z \mid \left|z\right|<1\right\}$ under the map $f(z) = z+z^{2}/2$.

Let

be an analytic function in the open unit disc $\mathbb {D}$. Assume that

Prove that $f$ is injective.

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 $z \in \mathbb {D}$.

For each $k>0$, let $X_{k}$ be the set of analytic functions $f(z)$ on the open unit disc $\mathbb {D}$ such that

is finite. Show that $f \in X_{k}$ if and only if $f' \in X_{k+1}$.

Let the function $f$ be analytic in the open unit disc of the complex plane and real valued on the radii $[0,1]$ and $[0, e^{i\pi \sqrt{2}}]$. Prove that $f$ is constant.

Let the function $f$ be analytic in the disc $\left|z\right|<1$ of the complex plane. Assume that there is a positive constant $M$ such that

Prove that

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?

Suppose that $f$ is analytic on the open upper half-plane and satisfies $\left|f(z)\right| \leq 1$ for all $z$, $f(i) = 0$. How large can $\left|f(2i)\right|$ be under these conditions?

Let $f(z)$ be analytic on the right half-plane $H = \left\{ z \mid \mathrm{Re}\left(z\right)>0\right\}$ and suppose $\left|f(z)\right| \leq 1$ for $z \in H$. Suppose also that $f(1) = 0$. What is the largest possible value of $\left|f'(1)\right|$?

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 $f$ be a holomorphic map of the unit disc $\mathbb {D} = \left\{ z \mid \left|z\right|<1\right\}$ into itself, which is not the identity map $f(z) = z$. Show that $f$ can have, at most, one fixed point.

If $f(z)$ is analytic in the open disc $\left|z\right|<1$, and $\left|f(z)\right| \leq 1/(1-\left|z\right|)$, show that

Let the function $f$ be analytic in the unit disc, with $\left|f(z)\right| \leq 1$ and $f(0) = 0$. Assume that there is a number $r$ in $(0,1)$ such that $f(r) = f(-r) = 0$. Prove that

Let $f(z)$ be an analytic function that maps the open disc $\left|z\right|<1$ into itself. Show that $\left|f'(z)\right| \leq 1/(1-\left|z\right|^{2})$.

Let $f$ be an analytic function in the open unit disc of the complex plane such that $\left|f(z)\right| \leq C/(1-\left|z\right|)$ for all $z$ in the disc, where $C$ is a positive constant. Prove that $\left|f'(z)\right| \leq 4C/(1-\left|z\right|)^{2}$.