Infinite Log-Concavity and r-Factor

TL;DR

This paper generalizes the concept of infinite log-concavity by introducing a new operator L_r, expanding the known regions of sequences that exhibit this property, and improving the criteria for log-concavity.

Contribution

It introduces a new operator L_r and redefines the hypersurfaces, extending the class of sequences proven to be infinitely log-concave under this generalized framework.

Findings

Defined the new operator L_r and the corresponding hypersurfaces.

Proved sequences within the new region are generalized r-factor infinitely log-concave.

Provided an improved value of r_0 for log-concavity criterion.

Abstract

D. Uminsky and K. Yeats [6] studied the properties of the log- operator L on the subset of the finite symmetric sequences and prove the existence of an infinite region R, bounded by parametrically de- fined hypersurfaces such that any sequence corresponding a point of R is infinitely log concave. We study the properties of a new operator L_r and redefine the hypersurfaces which generalizes the one defined by Uminsky and Yeats [6]. We show that any sequence corresponding a point of the region R, bounded by the new generalized parametrically defined r-factor hypersurfaces, is Generalized r-factor infinitely log concave. We also give an improved value of r_0 found by McNamara and Sagan [4] as the log-concavity criterion using the new log-operator.