  Hardy property of means has been extensively studied by P\'ales and Pasteczka
since 2016. The core of this research is based on few of their properties:
concavity, symmetry, monotonicity, repetition invariance and homogeneity (last
axiom was recently omitted using some homogenizations techniques). In the
present paper we deliver a study of possible omitting monotonicity and
replacing repetition invariance by a weaker axiom.
  These results are then used to establish the Hardy constant for certain types
of mixed means.
