  Let $S$ be a countable set provided with a partial order and a minimal
element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a
quasi-stationary distribution. We use Holley inequality to obtain sufficient
conditions under which the following hold. The trajectory of the chain starting
from the minimal state is stochastically dominated by the trajectory of the
chain starting from any probability on $S$, when both are conditioned to
nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to
this deterministic initial condition is the unique minimal quasi-stationary
distribution in the sense of stochastic order. As an application, we provide
new proofs to classical results in the field.
