  We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic
functions involving an operator having generalized Orlicz-growth embracing
besides Orlicz case also natural ranges of variable exponent and double-phase
cases. In particular, Harnack's Principle and Minimum Principle are provided
for $\mathcal{A}$-superharmonic functions and boundary Harnack inequality is
proven for $\mathcal{A}$-harmonic functions.
