On an integral inequality and related summability results
A class \(E^ q_ p(Q,C)\) of functions satisfying a certain reverse integral inequality is introduced. More precisely, a positive measurable function \(f\) belongs to the class \(E^ q_ p(Q,C)\), where \(C>1\), \(p,q\neq0\), \(p<q\), and \(Q\subset\mathbb R\) is an interval, if \(f\in L^ q(I)\) and \[ \frac1{| I| }\int_ If^ q(x)\,dx \leq C\big(\operatornamewithlimits{ess\;inf} _{x\in I}f(x)\big)^{q-p} \frac1{| I| }\int_ If^ p(x)\,dx \] for all intervals \(I\subset Q\). The author derives the relationship between \(E^ q_ p(Q,C)\) and the class \(B^ q_ p(Q,K)\) which contains measurable functions \(f\) satisfying the reverse Hölder inequality, \[ \Big(\frac1{| I| }\int_ If^ q(x)\,dx \Big)^{1/q} \leq K \Big( \frac1{| I| }\int_ If^ p(x)\,dx \Big)^{1/p}, \] for all intervals \(I\subset Q\). Moreover, relationships between \(B^ q_ p(Q,K)\) and the classical Gehring and Muckenhoupt classes are also discussed. The results of the paper give improvements of the results from \textit{L. Nania} [J. Aust. Math. Soc., Ser. A 49, No. 2, 319-326 (1990; Zbl 0715.26010)] and \textit{H. Alzer} [J. Math. Anal. Appl. 190, No. 3, 774-779 (1995; Zbl 0823.26013)].
scientific article; zbMATH DE number 2108637