  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.
