Concavity, Uniform Boundedness

Question

Can someone explain the inequality? What property of concave functions is this using?

Answer 1

Non-negative concave functions are subadditive, see for example Concave implies subadditive: $$ f(t+\tau) \le f(t) + f(\tau) \, . $$ For the estimate in the other direction use that $g(t) = f(1-t)$ is also concave: $$ f(t) = g(1-t) \le g(1-t-\tau) + g(\tau) = f(t+\tau) + f(1-\tau) \, . $$