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

Abstract

Let $G$ be a finite group and $H$ be a proper normal subgroup of $G$. In this note, we introduce two new functions $o(G,H)$ and $\psi''(G,H)$ involving the sums of element orders of $G$ and $H$. We prove that they satisfy certain inequalities, with equality if and only if $(G,H)$ is an equal order pair of a particular type.