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Volkmer observed that the inequality \\[(\\int_{-1}^1\\frac{1}{|r|}|f'|dx)^2 \\le K^2 \\int_{-1}^1|f|^2dx\\int_{-1}^1\\Big|\\Big(\\frac{1}{r}f'\\Big)'\\Big|^2dx \\] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \\[ -y\"=\\lambda\\, r(x)y,\\quad y(-1)=y(1)=0 \\] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$.\n  We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. 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