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I wonder if there is a general method for obtaining bounds on an analytic function using only its Taylor expansion (not using its special properties such as satisfying a good differential equation, etc.)

As a toy example, can we prove that $|sin(x)|\leq 1$ (or a weaker bound) only knowing that $sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$.

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    $\begingroup$ Note that if you make even a small perturbation to just one coefficient of the series of $\sin x$ the resulting series will be unbounded, so any criterium of boundedness should be sensible to this. $\endgroup$
    – Pietro Majer
    Dec 4, 2017 at 8:42
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    $\begingroup$ The Riemann hypothesis is equivalent to the behavior of a simple power series. So this is a difficult problem. In my paper arxiv.org/abs/1505.00440 I give a solution for a particular example of the problem. This can be applied to similar cases, but is not useful in the case of Riesz and Hardy and Littlewood series related to RH. $\endgroup$
    – juan
    Dec 4, 2017 at 13:49

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There is no such general method. You cannot see directly from the Taylor series that $\sin x$ is bounded on the real line, or that $\exp z$ is bounded on the negative ray. Of course what I stated is not a theorem, but just think how this boundedness criterion could possibly look: ANY change in ONE coefficient of the series destroys the property.

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