Bounding an index by the largest character degree of a solvable group

In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable group, then $\mathrm{\mid }G:O_p(G){\mathrm{\mid }}_p\le (b(G{)}^p\mathrm{/}p{)}^{1\mathrm{/}(p-1)}$ where $b(G)$ is the largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow $p$-subgroup of $G$ is at most $p$, then $\mathrm{\mid }G:O_p(G){\mathrm{\mid }}_p\le b(G)$.