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

Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities