
In One Line and Close. Permutations as Linear Orders. 23
THEOREM 1.32
If a sequence of positive real numbers has real roots only, then it is log-concave.
PROOF Let a
0
,a
1
, ···,a
n
be our sequence, and let P (x)=
n
k=0
a
k
x
k
.
Then for all roots (x, y) of the polynomial Q(x, y)=
n
k=0
a
k
x
k
y
n−k
,theratio
(x/y) must be real. (Otherwise x/y would be a non-real root of P (x)). There-
fore, by Rolle’s Theorem, this also holds for the partial derivatives ∂Q/∂x and
∂Q/∂y. Iterating this argument, we see that the polynomial ∂
a+b
Q/∂x
a
∂y
b
also has real zeros, if a + b ≤ n − 1. In particular, this is true in the special
case when a = j − 1, and b = n − j − 1, for some fixed ...