  It is well known that if Hardy-Littlewood maximal operator is bounded in
space $L^{p(\cdot)}[0;1]$ then $1/p(\cdot)\in BMO^{1/\log}$. On the other hand
if $p(\cdot)\in BMO^{1/\log},$ ($1<p_{-}\leq p_{+}<\infty$), then there exists
$c>0$ such that Hardy-Littlewood maximal operator is bounded in
$L^{p(\cdot)+c}[0;1].$ Also There exists exponent $p(\cdot)\in BMO^{1/\log},$
($1<p_{-}\leq p_{+}<\infty$) such that Hardy-Littlewood maximal operator is not
bounded in $L^{p(\cdot)}[0;1]$. In the present paper we construct exponent
$p(\cdot),$ $(1<p_{-}\leq p_{+}<\infty)$, $1/p(\cdot)\in BLO^{1/\log}$ such
that Hardy-Littlewood maximal operator is not bounded in $L^{p(\cdot)}[0;1]$.
