Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Question

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for
|z| < 1. Then,

(a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle;

(b) the set {z : |h'(z)| = 5} is a circle of strictly positive radius;

(c) h(1) = 10;

(d) regardless of what h' is, h'' ≡ 0.

A note to the reader: Actually I am preparing for an exam which is due next month. I know very little complex analysis and so I am trying to study only the key concepts. What I mean to say is I want to focus on topics/formulaes/concepts which might help me solve the above problem. So I would request you to mention the topics required in order to solve these questions.

My attempt: I know the cauchy riemann equations but using them here leads to complicated equations containing partial derivatives.

Answer 1

Hint: Apply Schwarz lemma to $f(z)/10$.