Shanin R. Reverse Jensen inequalities and functions mean oscillation

A relation between the equimeasurable rearrangement of function and the equimeasurable rearrangement of restriction of this function is obtained. We obtain the exact estimation of equimeasurable rearrangements of functions satisfying the reverse Holder inequality or the reverse Jensen inequality. An operator of extension of bounded mean oscillation functions from a segment to whole real line is constructed such that the extension has the asymptotically best norm. The John-Nirenberg inequality is extended to generalization BMO classes. In terms of the Orlich classes we proved some integral interrelations between generalized maximal functions by Hardy-Littlewood and by Fefferman-Stein.