Product maximizer on a compact convex subset of the positive orthant
= Product maximizer on a compact convex subset of the positive orthant
On the positive orthant,
$$
F(x)=\sum_{j=1}^n\log x_j
$$
is strictly concave. Its unique maximizer $x^*$ on a compact convex feasible set satisfies the first-order inequality
$$
\nabla F(x^*)\cdot(x-x^*)\leq0,
$$
equivalently
$$
\sum_{j=1}^n\frac{x_j}{x_j^*}\leq n.
$$