Two new functions related to the sum of element orders of a finite group

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] be a proper normal subgroup of [Formula: see text]. In this note, we introduce two new functions [Formula: see text] and [Formula: see text] involving the sums of element orders of [Formula: see text] and [Formula: see text]. We prove that they satisfy certain inequalities, with equality if and only if [Formula: see text] is an equal order pair of a particular type.