# UMD Analysis Qualifying Exam/Jan09 Complex

## Problem 4

 Suppose that $f,g\!\,$ are analytic on $\{|z|\leq 1\}\!\,$ with $g\neq 0\!\,$ on $\{|z|<1\}\!\,$ . Prove that $|f(z)|\leq |g(z)|\!\,$ for all $z\in \{|z|=1\}\!\,$ implies $|f(0)|\leq |g(0)|\!\,$ ## Solution 4

### Define new function h(z)

Define $h(z)={\frac {f(z)}{g(z)}}\!\,$ .

#### h is continuous on the closure of D

Since $g\neq 0\!\,$  on $D\!\,$ , then by the Maximum Modulus Principle, $g\!\,$  is not zero in ${\overline {D}}\!\,$ .

Hence, since $f\!\,$  and $g\!\,$  are analytic on ${\overline {D}}\!\,$  and $g\neq 0\!\,$  on ${\overline {D}}\!\,$ , then $h\!\,$  is analytic on ${\overline {D}}\!\,$  which implies $h\!\,$  is continuous on ${\overline {D}}\!\,$

#### h is analytic on D

This follows from above

### Case 1: h(z) non-constant on D

If $h\!\,$  is not constant on ${\overline {D}}\!\,$ , then by Maximum Modulus Principle, $|h|\!\,$  achieves its maximum value on the boundary of $D\!\,$ .

But since $|h(z)|\leq 1\!\,$  on $\partial D\!\,$  (by the hypothesis), then

$|h(z)|\leq 1\!\,$  on ${\overline {D}}\!\,$ .

In particular $|h(0)|\leq 1\!\,$ , or equivalently

$|f(0)|\leq |g(0)|\!\,$

### Case 2: h(z) constant on D

Suppose that $h(z)\!\,$  is constant. Then

$|h(z)|=\left|{\frac {f(z)}{g(z)}}\right|=|\alpha |\!\,$  where $\alpha \in \mathbb {C} \!\,$

Then from hypothesis we have for all $z\in \{|z|=1\}\!\,$ ,

$|f(z)|=|\alpha ||g(z)|\leq |g(z)|\ \!\,$

which implies

$|\alpha |\leq 1\!\,$

Hence, by maximum modulus principle, for all $z\in D\!\,$

$\left|{\frac {f(z)}{g(z)}}\right|=|\alpha |\leq 1\!\,$

i.e.

$|f(z)|\leq |g(z)|\!\,$

Since $0\in D\!\,$ , we also have

$|f(0)|\leq |g(0)|\!\,$