Mutual estimates of $L^p$-norms, and the Bellman function

A description of the possible values of the $L^p$-norm of a function is obtained under fixed $L^p$-norms for two other values of $p$ and under a natural multiplicative restriction, like the Muckenhoupt condition. Among special cases of our results, we mention simple interpolation inequalities between two $L^p$-norms, as well as nontrivial ones, such as the Gehring inequality or the reverse Hölder inequality for Mackenhoupt weights. The method of the paper is construction of the true Bellman function for the corresponding extremal problem. Bibl. – 5 titles.