  We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M.
Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics,
32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions}
under the usual (dominance) partial-order on partitions. We believe that our
proof technique may be helpful in obtaining similar inequalities for other
symmetric functions.
