Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment XIV.

1. Problem 1. a) Find isolated singularities and describe the behavior of the function near ∞:

ℵ) \frac{1}{e^{z} − 1} − \frac{1}{z};  ℶ) e^{z/(1−z)};  ℷ) sin (\frac{1}{sin(1/z)}).

b) Construct examples of functions having in the extended plane only the following singularities: the pole of the order m at 0 and the pole of the order n at ∞;

2. Problem 2. a) Determine explicitly the largest disk about the origin whose image under the mapping w = z^{2} + z is one to one.

b) Same problem for w = e^{z}.

c) Apply the representation f(z) = w_{0} + ξ(z)^{n} to cos z with z_{0} = 0. Determine ξ(z) explicitly.

d) If f(z) is analytic at the origin and f′(0) ≠ 0, prove the existence of an analytic g(z) such that f(z^{n}) = f(0) + g(z)^{n} in a neighborhood of 0.

3. Problem 3. a) Show that |f(z)| ≤ 1 for |z| ≤ 1 implies

\frac{|f′(z)|}{1 − |f(z)|^{2}} ≤ \frac{1}{1 − |z|^{2}} .

b) If f(z) is analytic and ℑf(z) ≥ 0 for ℑz > 0, show that

\frac{|f(z) − f(z_{0})|}{|f(z) − \overline{f(z_{0})}|} ≤ \frac{|z − z_{0}|}{|z − \overline{z}_{0}|}, \frac{|f′(z)|}{ℑf(z)} ≤ \frac{1}{y}.

c) In a) and b) prove that equality implies that f(z) is a fractional linear transformation.

d) Derive the corresponding inequalities if f(z) maps |z| < 1 into the upper half plane.

e) Prove by use of Schwarz’s lemma that every one-to-one conformal mapping of a disk onto another (or a half plane) is given by a fractional linear transformation.

4. Problem 4.

a) Let f be analytic in a square centered at the origin with sides, say, h_{1}, h_{2}, h_{3}, h_{4}. Assume that |f(z)| ≤ 1 ∀z ∈ h_{j}, j = 1, 2, 3, and |f(z)| ≤ 16 ∀z ∈ h_{4}. Give the best possible (in your opinion) upper bound for |f(0)|.

b) Let A be a set of functions f analytic in the strip |ℑz| ≤ π, and such that |f(±iπ)| ≤ 1. Is it possible to give an upper bound for sup_{f∈A} |f(0)|?

Hint: Consider e^{e^{z}}.