# UMD Analysis Qualifying Exam/Jan09 Complex

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## Problem 2[edit | edit source]

## Solution 2[edit | edit source]

## Problem 4[edit | edit source]

Suppose that are analytic on with on . Prove that for all implies |

## Solution 4[edit | edit source]

### Define new function h(z)[edit | edit source]

Define .

#### h is continuous on the closure of D[edit | edit source]

Since on , then by the Maximum Modulus Principle, is not zero in .

Hence, since and are analytic on and on , then is analytic on which implies is continuous on

#### h is analytic on D[edit | edit source]

This follows from above

### Case 1: h(z) non-constant on D[edit | edit source]

If is not constant on , then by Maximum Modulus Principle, achieves its maximum value on the boundary of .

But since on (by the hypothesis), then

on .

In particular , or equivalently

### Case 2: h(z) constant on D[edit | edit source]

Suppose that is constant. Then

where

Then from hypothesis we have for all ,

which implies

Hence, by maximum modulus principle, for all

i.e.

Since , we also have