The question is in the title. There are actually two variants, one for real analytic functions and the other one for complex analytic functions. It came up as an attempt to solve this question by analytic means, but I am not sure it really helps. I am afraid it is just a more difficult problem, unless there is a counterexample.
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2false for complex analytic case, counterexample, $f(x) = x^2 - \omega x$, $g(x) = -\omega x$ where $\omega$ is a primitive cubic root of unity. – achille hui Mar 05 '16 at 20:06