  Using a nonlinear version of the well known Hardy-Littlewood inequalities, we
derive new formulas for decreasing rearrangements of functions and sequences in
the context of convex functions. We use these formulas for deducing several
properties of the modular functionals defining the function and sequence spaces
$M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in
\cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in
a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it
decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general
setting, where they may even not be linear, and identify their K\"othe duals
with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of
rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to
be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the
class $M_{\varphi,w}$ is a separable quasi-Banach space,
$\mathcal{M}_{\varphi,w}$ is its Banach envelope.
