Math Problems: Complex Analysis

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Problem 11

11. Let f be entire with f(0)=1 and let r>0. Show that for some z with |z|=r we have e^(-z)f(z) in [1,+oo[.
Solution

Problem 12

12. f is analytic on the open disc D((0,0),1) such that limit |f(z)| -> 1 as |z| -> 1. Show that f(D) C D.
Solution

Problem 13

13. Suppose f and g are two entire functions such that |f(z)| <= |g(z)|. Prove that there exists c in C, f(z)=c.g(z).
Solution

Problem 14

14. g <> id is holomorphic on the open disc D((0,0),1). Show that g has at most one fixed point.
Solution

Problem 15

15. Are there any functions f and g analytic on the closed disk D((0,0),1) and such that:
(a) for all n in N, f(1/n)=f(-1/n)=1/n^2
(b) for all n in N, g(1/n)=g(-1/n)=1/n^3
Solution

Problem 16

16. Suppose h is entire, h(0)=3+4i and |h(z)| <= 5 if |z|<1. Find h'(0).
Solution

Problem 17

17. f is analytic and non identically zero on the closed disc D((0,0),1) such that |f(z)|<1 for |z|=1. Prove that the equation f(z)=z^3 admits 3 solutions (counting multiplicities) on the open unit disc.
Solution

Problem 18

18. a_0 >= a_1 >= ... >= a_n > 0. Show that the equation P(z)=a_0+a_1z+...+a_nz^n=0 has no zero in the open unit disc.
Solution

Problem 19

19. Suppose f is analytic on the open unit disc. Prove that there is a sequence (z_n) such that |z_n|<1, |z_n| -> 1 and (f(z_n)) is bounded.
Solution

Problem 20