Math Problems Vault

Problem 5.2.1

Find a one-to-one holomorphic map from the unit disk $\left\{ \left|z\right| < 1\right\}$ onto the slit disk $\left\{ \left|w\right| < 1\right\} - [0,1)$ .

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Problem 5.2.2

(a)(b)

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(a)

Find a function $f$ that conformally maps the region $\left\{ z : \left|\arg z\right| < 1\right\}$ one-to-one onto the region $\left\{ w : \left|w\right| < 1\right\}$ . Show that the function you have found satisfies the required conditions.

(b)

Is it possible to require that $f(1) = 0$ and $f(2) = \frac{1}{2}$ ? If yes, give an explicit map; if No, explain why not.

Problem 5.2.3

(a)(b)

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(a)

Find one 1-1 onto conformal map $f$ that sends the open quadrant $\left\{ (x,y) : x > 0 \text{ and } y > 0\right\}$ onto the open lower half disc $\left\{ (x,y) : x^2+y^2 < 1 \text{ and } y < 0\right\}$ .

(b)

Find all such $f$ .

Problem 5.2.4

Map the disk $\left\{ \left|z\right| < 1\right\}$ with slits along the segments $[a,1], [-1,-b]$ ( $0 < a < 1$ , $0 < b < 1$ ) conformally onto the full disk $\left\{ \left|w\right| < 1\right\}$ by means of a function $w = f(z)$ with $f(0) = 0$ , $f'(0) > 0$ . Compute $f'(0)$ and the lengths of the arcs corresponding to the slits.

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Problem 5.2.5

Let $0 < \varepsilon < \pi$ , let $\gamma_\varepsilon$ denote the arc $\left\{ e^{it} : \varepsilon \leq t \leq 2\pi -\varepsilon \right\}$ and let $\Omega_\varepsilon$ be the complement of $\gamma_\varepsilon$ in the Riemann sphere. If $f$ is the conformal map of the unit disk onto $\Omega_\varepsilon$ , $f(0) = 0$ , $f'(0) > 0$ , describe the part of the unit disc that $f$ maps onto $\left\{ \left|z\right| > 1\right\}$ .

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Problem 5.2.6

(a)(b)

Suppose that $w = f(z)$ maps a simply connected region $G$ one-to-one and conformally onto a circular disk $D_r$ with center $w = 0$ , radius $r$ , such that $f(a) = 0$ and $\left|f'(a)\right| = 1$ for some point $a \in G$ .

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(a)

Prove that the radius $r = r(G,a)$ of $D_r$ is uniquely determined by $G$ and $a$ .

(b)

Determine $r(G,a)$ if $G$ is the region between the hyperbola $xy = 1$ ( $x>0, y>0$ ) and the positive axes, and if $a = 1+\frac{i}{2}$ .

Problem 5.2.7

(a)(b)

Let $T(z) = \dfrac {az+b}{cz+d}$ be a Mobius transformation.

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(a)

Assume that $z_1, z_2 \in \mathbb {C}$ are two distinct fixed points for $T$ , i.e., $T(z_i) = z_i$ , $i=1,2$ . Show that there exists a constant $c$ such that

\frac{T(z)-z_1}{T(z)-z_2} = c\, \frac{z-z_1}{z-z_2}.

(b)

Use (a) to find an expression for $T^n(z)$ , $n=1,2,3,\ldots$ , if

T(z) = \frac{1-3z}{z-3}.

Problem 5.2.8

(a)(b)(c)

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(a)

Justify the statement that ``the curves

\frac{x^2}{a^2+\lambda } + \frac{y^2}{b^2+\lambda } = 1

form a family of confocal conics''.

(b)

Prove that such confocal conics intersect orthogonally, if at all.

(c)

Show that the transformation $w = \frac12\left(z+\frac1z\right)$ carries straight lines through the origin and circles centered at the origin into a family of confocal conics.

Problem 5.2.9

If $f : D(0,1) = \left\{ \left|z\right| < 1\right\} \to \mathbb {C}$ is an analytic function which satisfies $f(0) = 0$ , and if

\left|\mathrm{Re}\left(f(z)\right)\right| < 1 \quad \text{for all } z \in D(0,1),

prove that

\left|f'(0)\right| \leq \frac4\pi .

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Problem 5.2.10

Let $\Omega = \left\{ z \in \mathbb {C}; -1 < \im z < 1\right\}$ , and let $\mathcal F$ be the family of all analytic functions $f : \Omega \to \mathbb {C}$ such that $\left|f\right| < 1$ on $\Omega$ and $f(0) = 0$ . Find

\sup _{f \in \mathcal F} \left|f(1)\right|.

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Problem 5.2.11

Let $f$ be an analytic function on $D = \left\{ z; \left|z\right| < 1\right\}$ such that $f(0) = 1$ , and suppose that $\left|1+f(z)\right| \leq 1 + \left|f(z)\right|$ whenever $\left|z\right| < 1$ . Prove that $\left|f'(0)\right| \leq 4$ .

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Problem 5.2.12

Let $P$ be the set of holomorphic functions $f$ on the open unit disc so that (i) Both the real and imaginary parts of $f(z)$ are positive for $\left|z\right| < 1$ , (ii) $f(0) = 1+i$ . Let $E = \left\{ f(\frac12) : f \in P\right\}$ . Describe $E$ explicitly.

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Problem 5.2.13

Let $\Omega = \left\{ z : \left|z-4\right| < 4\right\}$ . If $\mathcal F$ is the family of all analytic functions on $\Omega$ such that $\left|f\right| < 1$ in $\Omega$ and $\lim_{w \to \infty } f(w) = 0$ , find $\sup_{f \in \mathcal F} \left|f(8)\right|$ . Your answer should be an explicit number, and you should prove your assertion.

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Problem 5.2.14

Let $D$ be the upper half plane and let $f \neq \mathrm{id}$ be a conformal map of $D$ onto itself such that $f \circ f = \mathrm{id}$ . Prove that $f$ has a unique fixed point inside $D$ .

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Problem 5.2.15

Let $\Omega$ be a convex, open subset of $\mathbb {C}$ and let $f : \Omega \to \mathbb {C}$ be an analytic function satisfying $\mathrm{Re}\left(f'(z)\right) > 0$ , $z \in \Omega$ . Prove that $f$ is one-to-one in $\Omega$ (i.e., $f$ is injective).

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Problem 5.2.16

Show that if the polynomial $P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ , $n > 1$ , is one-to-one in the unit disk $\left|z\right| < 1$ and $a_1 = 1$ , then $\left|na_n\right| \leq 1$ .

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Problem 5.2.17

(a)(b)

Let $P(z)$ be a polynomial on the complex plane, not identically zero; let $H = \left\{ z : \re z > 0\right\}$ .

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(a)

If all roots of $P(z)$ lie in $H$ , show that the same is true for the roots of $dP/dz$ .

(b)

For any non-vanishing polynomial $P(z)$ , use the result in (a) to show that the convex hull of the roots of $P(z)$ contains the roots of $dP/dz$ .

Problem 5.2.18

(a)(b)

Let $f(z)$ be a Laurent series centered at $0$ , convergent in $\mathbb {C} \setminus \left\{ 0\right\}$ , with residue $b$ at $z=0$ .

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(a)

Show that there exists $\zeta$ on $\left\{ z \in \mathbb {C} : \left|z\right| = 1\right\}$ with

\left|f(\zeta ) - \zeta ^{-1}\right| \geq \left|b-1\right|.

(b)

Characterize those functions with

\max _{\left|\zeta \right|=1} \left|f(\zeta )-\zeta ^{-1}\right| = \left|b-1\right|.

Problem 5.2.19

(a)(b)(c)

Assume $f$ is analytic in a neighborhood of $\closure D$ , $f$ maps $D$ into $D$ , and $f$ maps $\partial D$ into $\partial D$ , where $D = \left\{ z : \left|z\right| < 1\right\}$ .

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(a)

Show that $\forall z \in \partial D$ , $f'(z) \neq 0$ .

(b)

Show that $\dfrac {d}{d\theta }[\arg f(e^{i\theta })] \geq 0$ for $\theta$ in $\mathbb {R}$ .

(c)

Assume that $f(0) = f'(0) = 0$ and $f|_{\partial D}$ is a two-to-one map from $\partial D$ onto $\partial D$ . Show that $f(z) \neq 0$ whenever $0 < \left|z\right| < 1$ .

Problem 5.2.20

If $f(z)$ is a univalent (1-1 analytic) function with domain the unit disc such that $f(z) = z\sum_{n=2}^{\infty } a_n z^n$ , then prove that

g(z) = \sqrt{f(z^2)}

is a univalent function in the unit disc.

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Problem 5.2.21

Consider the unit disk $D$ and an analytic function $f : D \to D$ . If $f(0) = a \neq 0$ , show that $f$ does not have a zero in the disk $\left\{ z : \left|z\right| < \left|a\right|\right\}$ .

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Problem 5.2.22

(a)(b)

Let $f(z) = \dfrac {z}{(1+z)^2}$ . Let $V = \left\{ z \in \mathbb {C} : 0 < \left|z\right| < \frac12, \frac{3}{4}\pi < \arg z < \frac54\pi \right\}$ .

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(a)

Show that if $z \in V$ , then $\left|f(z)\right| < \left|z\right|$ and $f(z) \in V$ .

(b)

Prove that if $z \in V$ , then $\lim_{n \to \infty } f^{(n)}(z) = 0$ , where $f^{(n)}(z) = f \circ f \circ \cdots \circ f$ is the $n$ -fold composition of $f$ .