Let $\{X_n, -\ity < n < \ity \}$ be a stationary $\phi$-mixing process with the one-dimensional marginal distribution function $F$ and the density function $f.$ Let $F_n(x)$ be the empirical distribution function based on the observations $\{X_i, 1 \leq i \leq n\}$ and $W_n^*= \sup_{-\ity < x < \ity}\sqrt n|F_n(x)-F(x)|.$ We obtain upper bounds for $E(W_n^*)$. We give an application to get bounds on the expectation of the supremum of the deviation of a kernel density estimator $\hat f_n(x)$ from true density function $f(x)$ . Similar results were obtained for a kernel type estimator $\hat F_n(x)$ for the true distribution function $F(x).$