I'm studying lecture notes on "Diffusion equations and entropy inequalities" http://mate.unipv.it/toscani/publi/Note-Ravello-2016.pdf
(1) The author asserts that (1.5) $$ N(X+Z_t)\ge(1-t)N(X)+tN(X+Z_1), \quad 0\le t\le1. $$ is stronger than (1.4) $$ N(X+Y)\ge N(X)+N(Y). $$
(2) (1.5) is equivalent to $$ \frac{d^2}{dt^2}N(X+Z_t)\le0 $$
I have looked for concave functions on wiki (https://en.wikipedia.org/wiki/Concave_function), but cannot see why the two facts are true. I Any reference, suggestion, idea, or comment is welcome. Thank you!