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 $sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$.



Proving bounds on analytic functions using only the Taylor expansion
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. 


