On some reverse integral inequalities (Q1270689)

Let \(a\in (0,\infty)\) be fixed. For \(\gamma, q\in(1,\infty)\) define \(M(\gamma, q)\) as the set of all positive functions on \((0,a]\) satisfying \[ f^q(t)\leq t^{-1} \int^t_0 f^q(x)dx\leq \gamma f^q(t),\quad t\in(0, a]. \] The main result of the paper states that \(M(\gamma, q)\subset L^p(0,a)\) for any \(p\in [q,q\gamma/(\gamma- 1))\) and that for all \(f\in M(\gamma, q)\), \[ \Biggl({1\over a} \int^a_0 f^p(x)dx\Biggr)^{1/p}\leq c\Biggl({1\over a} \int^a_0 f^q(x)dx\Biggr)^{1/q}, \] where \(c= [\gamma^{r+ 1}/(\gamma- r(\gamma- 1))]^{1/p}\) and \(r= p/q\).