  We introduce a notion of $F$-concavity which largely generalizes the usual
concavity. By the use of the notions of closedness under positive scalar
multiplication and closedness under positive exponentiation we characterize
power concavity and power log-concavity among nontrivial $F$-concavities,
respectively. In particular, we have a characterization of log-concavity as the
only $F$-concavity which is closed both under positive scalar multiplication
and positive exponentiation. Furthermore, we discuss the strongest
$F$-concavity preserved by the Dirichlet heat flow, characterizing
log-concavity also in this connection.
