We investigate the log-concavity on the half-line of the Wright function $φ(-α,β,-x),$ in the probabilistic setting $α\in (0,1)$ and $β\ge 0.$ Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for $β\geqα$ and in the classical case $β= 1-α$ of the Mittag-Leffler distribution, which exhibits a certain critical parameter $α_*= 0.771667...$ defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if $β\geqα$ or $α\le 1/2$ and $β= 0.$