Generalization of a theorem on the quasi-analyticity of a function

It is a known fact that if a function, together with all of its derivatives, vanishes at a point, then the function will be zero in a neighborhood of the point if its successive derivatives satisfy certain estimates. We show that even if the function does not have a priori all of its derivatives but is such that its first derivative has a special sequence of majorizing functions, then in this case also the function will be equal to zero. We use our results to obtain theorems concerning the uniqueness of the solution of an abstract Cauchy problem.