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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 $L^{p(\\cdot)}[0;1]$. 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