Let $x(t)$ be a random function with known ${\bf E}[x(t)]$ and ${\bf E}[x(t)x(s)]$, $0 \leqq s$, $t \leqq 1$. In Section 3 a bound is given for the probability that $|x(t)|$ exceeds the given function $\alpha (t)$ at least for one t. The bound involves an arbitrary quadratic form, which can be selected in an appropriate way giving certain bounds (see, for example, formula (30)). The effectiveness of this method depends on the degree of differentiability of $x(t)$. In the last two sections the case is treated when x is a function of several variables $t_1 ,t_2 , \cdots ,t_m $.