The largest character degrees of the symmetric and alternating groups

We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{\psi\in\textrm{Irr}(A_n),\,\psi(1)< b(A_n)}\psi(1)^2, \] where $\textrm{Irr}(A_n)$ and $b(A_n)$ respectively denote the set of irreducible complex characters of $A_n$ and the largest degree of a character in $\textrm{Irr}(A_n)$. This confirms a prediction of I. M. Isaacs for the alternating groups and answers a question of M. Larsen, G. Malle, and P. H. Tiep.