Characterizations of monotone O-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 KreÄ­n-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 KreÄ­n-Feller eigenvalue problems allow a similar characterization of the class of nondecreasing O-regularly varying functions.