  In this paper we present several new classes of logarithmically completely
monotonic functions. Our functions have in common that they are defined in
terms of the $q-$gamma and $q-$digamma functions. As an applications of this
results, some inequalities for the $q-$gamma and the $q-$digamma functions are
established. Some of the given results generalized theorems due to Alzer and
Berg and C.-P. Chen and F. Qi.
