Characterizations of monotone -regularly varying functions by means of indefinite eigenvalue problems and HELP type inequalities

Connections between different areas of mathematical analysis are obtained: regular variation, indefinite operator theory and HELP type inequalities. Using a result by Parfyonov, the nondecreasing and so-called positively increasing functions induce precisely those indefinite Kreĭn–Feller eigenvalue problems such that the eigenfunctions have the Riesz basis property. Furthermore, these functions induce precisely those Lebesgue–Stieltjes measures such that the associated HELP type inequalities are valid. The so-called dual Kreĭn–Feller eigenvalue problems allow a similar characterization of the class of nondecreasing O -regularly varying functions.