Question 4
Let $n$ be a positive integer and define
$$
f(j) = \binom{2j}{j} \cdot \binom{2n-2j}{n-j}, \quad j = 0, 1, \dots, n.
$$
Prove that
(a) $f(j) \ge f(j+1)$, if $0 \le j \lt n/2$;
(b) $f$ is strictly convex, i.e.,
$$
f(j+1) + f(j-1) \gt 2f(j), \quad j = 1, 2, \dots, n-1.
$$
$$
f(j) = \binom{2j}{j} \cdot \binom{2n-2j}{n-j}, \quad j = 0, 1, \dots, n.
$$
Prove that
(a) $f(j) \ge f(j+1)$, if $0 \le j \lt n/2$;
(b) $f$ is strictly convex, i.e.,
$$
f(j+1) + f(j-1) \gt 2f(j), \quad j = 1, 2, \dots, n-1.
$$