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Problems Home

**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**.

Solution

**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

**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

**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

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

Solution

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

Solution

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

Solution

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

Solution

**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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