Generalization of two results of the theory of uniform distribution

For a sequence ${x_1}, \ldots ,{x_N}$ of points in $[0,1]$ and a sequence ${p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)$ of nonnegative numbers, define the distribution function

$$g(x) = x - \sum\limits_{{x_k} < x} {{p_k}.}$$

Let $\varphi$ be an increasing function on $[0,1]$ and $\varphi (0) = 0$. The main result of the paper is

$$F({D_N}) \leqslant \int_0^1 {\varphi \left( {\left\vert {g(x)} \right\vert} \right)} dx \leqslant \varphi ({D_N}),$$

where ${D_N}$ is the supremum norm of $g$ on $[0,1]$ and $F$ is the antiderivative of $\varphi$ with $F(0) = 0$. This result generalizes and improves an estimate of Niederreiter [1] for the ${L^2}$ discrepancy of the sequence ${x_1}, \ldots ,{x_N}$. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.