  We study the question of how much one can weaken the defining condition of
BMO. Specifically, we show that if $Q$ is a cube in $\mathbb{R}^n$ and
$h:[0,\infty)\to[0,\infty)$ is such that
$h(t)\underset{t\to\infty}{\longrightarrow}\infty,$ then
  $$ \sup_{J \text{subcube} Q} \frac1{|J|}\int_J h(|\varphi-\frac1{|J|}
\int_J\varphi |)<\infty \Longrightarrow \varphi\in BMO(Q). $$
  Under some additional assumptions on $h$ we obtain estimates on
$\|\varphi\|_{BMO}$ in terms of the supremum above. We also show that even
though the condition $h(t)\underset{t\to\infty}{\longrightarrow}\infty$ is not
necessary for this implication to hold, it becomes necessary if one considers
the dyadic BMO.
