  It was recently proven that the correlation function of the stationary
version of a reflected L\'evy process is nonnegative, nonincreasing and convex.
In another branch of the literature it was established that the mean value of
the reflected process starting from zero is nonnegative, nondecreasing and
concave. In the present paper it is shown, by putting them in a common
framework, that these results extend to substantially more general settings.
Indeed, instead of reflected L\'evy processes, we consider a class of more
general stochastically monotone Markov processes. In this setup we show
monotonicity results associated with a supermodular function of two coordinates
of our Markov process, from which the above-mentioned monotonicity and
convexity/concavity results directly follow, but now for the class of Markov
processes considered rather than just reflected L\'evy processes. In addition,
various results for the transient case (when the Markov process is not in
stationarity) are provided. The conditions imposed are natural, in that they
are satisfied by various frequently used Markovian models, as illustrated by a
series of examples.
