  We present a class of modified logarithmic Sobolev inequality, interpolating
between Poincar\'e and logarithmic Sobolev inequalities, suitable for measures
of the type $\exp(-|x|^\al)$ or more complex $\exp(-|x|^\al\log^\beta(2+|x|))$
($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities.
These modified inequalities share common properties with usual logarithmic
Sobolev inequalities, as tensorisation or perturbation, and imply as well
Poincar\'e inequality. We also study the link between these new modified
logarithmic Sobolev inequalities and transportation inequalities.
