A fourth derivative test for exponential sums

We give an upper bound for the exponential ${\sum }_{m=1}^M\exp (2i\pi f(m))$ in terms of $M$ and $\lambda$, where $\lambda$ is a small positive number which denotes the size of the fourth derivative of the real valued function $f$. The classical van der Corput’s exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.