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A graduate level question in complex analysis

  1. Jul 24, 2008 #1
    If f and g are two entire functions such that mod(f(z)) <= mod(g(z)) for all z in C, prove that f=cg for some complex constant c.
     
  2. jcsd
  3. Jul 24, 2008 #2
    I tried to prove this by applying Liouville's theorem to f/g.
    it is clear that f/g is bounded.
    and f/g is analytic except at the zeros of g.
    if z1 is a zero of g its a zero of f also.
    and if multiplicity of z1 as a zero of g will be less than or equal to multiplicity of z1 as a zero of f.
    i got stuck here.
    pls help me to prove f/g is analytic at zeros of g also.
    so that by Liouville's theorem f/g will be a constant.
     
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