Two analogs of Pleijel's inequality

TL;DR

This paper develops two analogs of Pleijel's inequality to recover distribution function properties from generalized Stieltjes transforms, extending classical inversion formulas to broader contexts.

Contribution

It introduces novel analogs of Pleijel's inequality for Riesz means and distribution functions with power growth, using generalized Stieltjes transforms.

Findings

Derived approximate inversion formulas for Riesz means from Stieltjes transforms.

Extended inversion techniques to distribution functions with power growth.

Presented a new theorem combining the two inversion scenarios.

Abstract

Pleijel's inequality is an approximate inversion formula for the Stieltjes transform (or Cauchy integral) of a distribution function on positive semi-axis. It implies a Tauberian theorem due to Malliavin. The proposed analogs of Pleijel's inequality deal with (1) approximate recovery of the Riesz means of the distribution function from its Stieltjes transform, and (2) approximate recovery of the distribution function with power growth for which the ordinary Stieltjes transform does not exist. In the latter case, a power of the Cauchy (or Stieltjes) kernel is used to define the "generalized Stieltjes transform". A previously unpublished theorem stated in Appendix pertains to combination of the two situations (input: generalized Stieltjes transform; output: Riesz means).