A criterion for $p$-solvability of finite groups, where $p=7$ or $11

For a finite group $G$, define $ \psi^{\prime \prime}(G)=\psi(G)/|G|^2 $, where $\psi(G)=\sum_{g\in G}o(g)$ and $o(g)$ denotes the order of $g \in G $. In this paper, we give a criterion for $p$-solvability by the function $\psi''$, where $ p \in \{7, 11\} $. We prove that if $ G $ is a finite group and $\psi''(G)>\psi''({\rm PSL}(2, p))$, where $p \in \{7, 11\}$, then $G$ is a $p$-solvable group.