sorry this is quite too long. clear handwriting plz. thank you Suppose f is holo

Question

sorry this is quite too long. clear handwriting plz. thank you

Suppose f is holomorphic in a neighbourhood of B(0, 1) (a) By the Maximum Principle, fl attains its maximum on B(0, 1) on the boundary. Show that either Ifl attains its minimum on B(0,1) or f has a zero in B(0, 1) (b) Now suppose If(z) 1 for |z 1. Show that either f is constant or f has a zero in B(0, 1). (c) Assume f is not constant. Briefly explain why f has a finite number of zeros ai, a2 , an in B(0, 1) (not necessarily distinct). (d) Recall that if a

Explanation / Answer

a) Consider the function g=1/f

Suppose f doesn't have a zero in B(0,1) then g is analytic in B(0,1) then by maximum modulus theorem |g| has a maximum on the boundary of B(0,1) which will imply that |f| has a minimum on the boundary

b) Consider the same function g

Suppose f is not constant

Assume f has no zero in B(0,1).

By max modulus thm |f(z)| <1 which will imply |g(z)| >1 .This is a contradiction to Max. modulus thm, since g is analytic and |g(z)|=1 for all |z|=1

c) If there are infinitely many zeros the function will become zero.

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