  We investigate the log-concavity on the half-line of the Wright function
$\phi(-\alpha,\beta,-x),$ in the probabilistic setting $\alpha\in (0,1)$ and
$\beta \ge 0.$ 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 $\beta\geq\alpha$ and in the classical case $\beta = 1-\alpha$ of
the Mittag-Leffler distribution, which exhibits a certain critical parameter
$\alpha_*= 0.771667...$ 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 $\beta\geq\alpha$ or $\alpha\le 1/2$ and $\beta = 0.$
