Inverse Hölder inequalities with weight \(t^ \alpha\) (Q1260850)

This interesting paper extends results of \textit{R. W. Barnard} and \textit{J. Wells} [J. Math. Anal. Appl. 147, No. 1, 198-213 (1990; Zbl 0731.26014)] on inverse Hölder inequalities with power weights for positive concave functions. For a pair of such functions \(u\), \(v\) on \([0,1]\) and for \(\alpha>1\) the following inequality is established using a minimization procedure: \[ \int_ 0^ 1 u(t)v(t)t^ \alpha dt\geq C_ \alpha(p)\| u\|_{p,t^ \alpha}\| v\|_{p',t^ \alpha}. \] The size of the best constant is discussed as well.