MATH502_HW6 - Serge Ballif MATH 502 Homework 6 February 29,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Serge Ballif MATH 502 Homework 6 February 29, 2008 (1) The function f ( z ) is entire, and it is known that | f ( z ) | 1 whenever | z | = 1 and that | f ( z ) | 10 whenever | z | = 10 . Show that | f ( z ) | 5 whenever | z | = 5 . What can be said about | f (0) | ? Claim 1. For all 1 | z | 10 we have | f ( z ) | | z | . Proof. Consider the function f ( z ) /z defined on A = { 1 < | z | < 10 } . This region is a connected open set on which f ( z ) /z is holomorphic (since no poles lie in the domain). Hence, we can invoke the maximum principle to see that | f ( z ) /z | attains its maximum on the the boundary A = {| z | = 1 , 10 } . By assumption we know that | f ( z ) /z | 1 on A . Thus, for every z A , we must have | f ( z ) /z | 1 or equivalently | f ( z ) | | z | . In particular, this shows that | f ( z ) | < 5 for | z | = 5. The only thing that we can conclude about f (0) is that it lies somewhere in the unit disc, because for any u...
View Full Document

Page1 / 2

MATH502_HW6 - Serge Ballif MATH 502 Homework 6 February 29,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online