I am trying to bound a function $ f(\mathbf{x},\mathbf{y})$ defined for $ 0\leq x_i \leq 1$ and $ y_i\geq 0$ , with $ \mathbf{x},\mathbf{y}\in\mathbb{R}^n$ . At present, I can prove that for all subsets $ S\subseteq\{1,\dots,n\}$ , this function satisfies $ $ \left(\prod_{i\in S} \left(x_i^{x_i}(1-x_i)^{1-x_i}\right)^{y_i}\right)^\frac{1}{\sum_{i\in S} x_i y_i} \sum_{i\in S} x_i y_i \leq f(\mathbf{x},\mathbf{y})\leq C\sum_{i=1}^nx_i^2 y_i$ $ for some constant $ C$ , and my question is: is there anything useful I can say about the left-hand expression? It doesn’t appear that these two expressions are within a constant factor of each other, but when I do simulations with random numbers, the gap between them does not seem huge.