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I am stuck on the following problem that says:

Let $a(z),b(z)$ be two non-zero complex polynomials. Then $a(z)\overline{b(z)}$ is analytic iff

  1. $a(z)$ is constant

  2. $a(z)b(z)$ is constant

  3. $\overline{a(z)}b(z)$ is constant

  4. $b(z)$ is constant

I am not sure how to tackle it. Can someone explain? Thanks in advance for your time.

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Try showing that if $f$ is analytic and nonconstant, then $\overline f$ is not analytic. –  Lubin Jun 27 '13 at 4:06
    
I think $a(z)$ is constant? –  Bunuelian Trick Jun 30 '13 at 3:49
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