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

22. Let f be an entire function such that f^(-1)(A) is bounded whenever A is bounded. Show that f(C)=C.
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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.
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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).
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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.
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26. f is analytic from the open unit disc into itself. Show that |f '(z)|<= 1/(1-|z|^2).
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27. Is there an entire function f satisfying: for all n in N-{0}, f(n)=n/(n+1)?
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28. Is there an entire function f satisfying: for all n in N-{0}, f(1/n)=n/(n+1) ?
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29. For all polynomial p with complex coefficients, there exists z in C, |z|=1 and |p(z)-1/z| >= 1.
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30. If Omega is a root of the polynomial P(z)=5z^4+z^3+z^2-7z+14, show that |Omega| <= 2.
Solution