I kown of two Vapnik-Chervonenkis inequalities. The original one, by Vapnik and Chervonenkis,
$$P\left(\sup_{A \in \mathcal{A}} \left |\nu_n(A) - \nu(A) \right | >\varepsilon \right) \leq 8\, S(\mathcal{A},n)\, e^{-n\varepsilon^2/32}$$
and the following (I am not sure whom to credit for - I read it in Devroye,Lugosi Combinatorial methods in density estimation)
$$\mathbb{E}\left[\sup_{A \in \mathcal{A}} \left |\nu_n(A) - \nu(A) \right | \right] \leq 2 \sqrt{\dfrac{\log 2\,S(\mathcal{A},n) + \log 2}{n}}$$
How do they compare?