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Problem 21
21. If a and b have nonpositive real parts, show that |e^a-e^b| <= |a-b|.
Solution

Problem 22
22. Let f be an entire function such that f^(-1)(A) is bounded whenever A is bounded. Show that f(C)=C.
Solution

Problem 23
23. Let f be a function analytic on C - J= {z; Im(z)<> 0} and continuous on C. Prove that f is analytic on C.
Solution

Problem 24
24. Find the radius of convergence of the Taylor series about z=1 of the function f(z)=1/(1+z^2+z^4+z^6+z^8+z^10).
Solution

Problem 25
25. f is analytic in the open unit disc D and takes real values on [0,1[ U [0,e^(i pi sqrt(2))[. Show that f is constant.
Solution


Problem 26
26. f is analytic from the open unit disc into itself. Show that |f '(z)|<= 1/(1-|z|^2).
Solution

Problem 27
27. Is there an entire function f satisfying: for all n in N-{0}, f(n)=n/(n+1)?
Solution

Problem 28
28. Is there an entire function f satisfying: for all n in N-{0}, f(1/n)=n/(n+1) ?
Solution

Problem 29
29. For all polynomial p with complex coefficients, there exists z in C, |z|=1 and |p(z)-1/z| >= 1.
Solution

Problem 30
30. If Omega is a root of the polynomial P(z)=5z^4+z^3+z^2-7z+14, show that |Omega| <= 2.
Solution


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