  A new bound for the remainder term in the Taylor expansion of the complex
exponent $e^{ix}$, $x\in\R$, is proved yielding precise moment-type estimates
of the accuracy of the approximation of the characteristic function (the
Fourier--Stieltjes transform) of a probability distribution by the first terms
of its Taylor expansion. Moreover, a precise upper bound for the third moment
of a probability distribution in terms of the absolute third moment is
established which sharpens Jensen's inequality. Based on these results, new
improved bounds for characteristic functions and their derivatives are obtained
that are {\it uniform} in the class of distributions with fixed first three
moments.
