Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic.
How does one prove this?
Surely, if Cauchy's estimates hold for the derivatives of a function, then its Taylor series converges, but that is not enough for analyticity.
Is it necessary to consider the smooth function on a compact interval? In the second answer to this question, the above fact is used for a function defined on the whole real line.