Question

D. (a) Show that if f is Let f be complex differentiable on a bounded domain D and is continuous on the closure D = DU non-zero on D the modulus $(2) attains it's minimum on the boundary aD. Hint: Consider FT2) (b) Give an example that shows that the assumption that f is non-zero on D is necessary.

Answer 1

Pr Recall some basic fact. Maximum Modulus theorem: If fiz) is analytic in a bounded domain D and continuous on its closure D then Ifra) attains a maximum on the boundary. Furthermore Ifra) does not attain a maximum at an interior point unless fiz) is constant. Now the given problemy Let f be complex differentiable on a bounded domain D. and is continuous on the closure D = DuaD @ we shall Show that if f is non zero on D the modules attain its minimum on the boundary 2D. the theosam I took If f vanishes at a point ad, then obviously true. Assume that fiz) to on a D. Then I is analytic on D and continuous on D. By na Maximum modulus theorem cannot attain its maximum value inside ad. and fence If(2) can not assume its minimum value inside aD. Since f(2) I has a minimum on D as f is continuous on a closed and bounded set i 1. this minimum must be attained or OD if f(2)=0 and D={ZEC : |Z| <d) with r: 121=4, en f(0) = 0 . IF 2€ (77, then I fx)) = 1 on 2D. which shows that assumption that is non-zero on D is necessary.