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Interpolation inequalities with power weights for functions of one variable. (English) Zbl 0779.26012

The aim of the paper is to find simple, sufficient conditions for the validity of an interpolation inequality of the form \[ \left(\int_ I| x^ \beta u^{(j)}|^ pdx\right)^{1/p}\leq K\left(\int_ I| x^ \gamma u|^ qdx\right)^{(1-\lambda)/q}\left(\int_ I| x^ \alpha u^{(n)}|^ rdx\right)^{\lambda/r}, \tag{1} \] where \(1\leq p,q,r\leq\infty\), \(0\leq j\leq n-1\), \(I\) is one of the intervals \((0,\infty)\), \((0,b]\) or \([b,\infty)\) with \(b>0\), and the function \(u\) is in an appropriate subspace of \(AC^{(n)}(I)\). The method used to prove the inequality (1) is elegant and it is based on some results on Hardy- type inequalities and the Gabushin inequality.
Reviewer: B.Opic (Praha)

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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