I will discuss the connections between weak transportation inequalities with quadratic-linear cost functions, Bobkov-Ledoux type modified log-Sobolev inequalities for convex/concave functions and the convex Poincare inequality. If time permits I will also mention some self normalized moment inequalities for general convex functions, which can be easily obtained from concentration estimates for Lipschitz convex functions. Based on joint work with Michal Strzelecki.