Comparison between lower and upper $ \alpha-$densities and lower and upper $ \alpha-$analytic densities

Let $\alpha$ be a real number, with $\alpha \geq - 1$. We prove a general inequality between the upper (resp. lower) $\alpha-$analytic density and the upper (resp. lower)$\alpha-$density of a subset $A$ of $\mathbb N^*$ (Proposition 2.1). Moreover, we prove by an example that the upper and the lower $\alpha$--densities and the lower and upper $\alpha$--analytic densities of $A$ do not coincide in general ({\it i.e.}, the inequalities proved in (2.1) may be strict). On the other hand, we identify a class of subsets of $\mathbb N^*$ for which these values do coincide in the case $\alpha > -1$.