  We provide a mild sufficient condition for a probability measure on the real
line to satisfy a modified log-Sobolev inequality for convex functions,
interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux
type inequality. As a consequence we obtain dimension-free two-level
concentration results for convex function of independent random variables with
sufficiently regular tail decay. We also provide a link between modified
log-Sobolev inequalities for convex functions and weak transport-entropy
inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.
