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[.
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.
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).
14. g <> id is holomorphic on the open disc D((0,0),1). Show that g has at most one fixed
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
16. Suppose h is entire, h(0)=3+4i and |h(z)| <= 5 if |z|<1. Find h'(0).
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.
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.
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.
20. Find all the analytic functions f defined on the unit disk and satisfying:
f ''(1/n)+f(1/n)=0 for every natural integer n >= 2.
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