This paper is concerned with the following question: if a characteristic function satisfies a certain property at the origin, what can be said about its behavior on the entire real line?

Ratan Dasgupta, Rates of Convergence to Normality for Some Variables with Entire Characteristic Function, Sankhyā: The Indian Journal of Statistics, Series A (1961-2002), Vol. 54, No. 2 (Jun., 1992), ...