Convexity of a volume-sampling error sequence

For positive numbers $\lambda_1,\ldots,\lambda_n$, let $e_j$ be their elementary symmetric polynomials, with $e_0=1$ and $e_{n+1}=0$. We prove that $(j+1)e_{j+1}/e_j$ is a decreasing convex sequence on $\{0,\ldots,n\}$. The proof reduces each second difference to the nonnegative quantity $s_1s_3-s_2^2$ for the reciprocal roots of a derivative of the generating polynomial. This proves Conjecture 1 of Dereziński, Khanna, and Mahoney. We give a self-contained application to fixed-size determinantal sampling, remove the concentration loss from its stable-rank bound, and derive a slightly sharper explicit bound.