A graduate level question in complex analysis

[vijigeeths]

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.

 

[vijigeeths]

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.