  We study one-dimensional functional inequalities of the type of Poincar\'e,
logarithmic Sobolev and Wirtinger, with weight, for probability densities with
polynomial tails. As main examples, we obtain sharp inequalities satisfied by
inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities,
taking values on $R$. In this last case, we improve the result obtained by
Bobkov and Ledoux in 2009 by introducing a better weight function in the
logarithmic Sobolev inequality. The results are obtained by resorting to
Fokker-Planck type equations which possess these densities as steady states.
