We give a necessary and sucient condition for transport entropy inequalities in dimension one. As an application, we construct a new example of a probability distribution verifying Talagrand's $T_2$ inequality and not the logarithmic Sobolev inequality.
"Transport-Entropy inequalities on the line." Electron. J. Probab. 17 1 - 18, 2012. https://doi.org/10.1214/EJP.v17-1864