**Question.** Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues 
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \lambda_n$ in terms of the quantity
$$
\|A\|_\infty := \max_{1 \leq i, j \leq n} |a_{ij}| ?
$$

A classic inequality due to A. Hirsch states that the modulus of an eigenvalue of an $n \times n$ complex matrix $A$ is less than $n  \|A\|_\infty$, which implies that 
$$
|\lambda_2 \cdots \lambda_n| \leq n^{n-1}\|A\|_\infty^{n-1} .
$$
However, this seems like a rather rough estimate for positive-definite matrices. I'm interested in any estimate that is substantially better than this.

**Motivation.** Using Hirsch's inequality one can improve Lemma 4 in page 43 of Siegel's [Lectures on Quadratic Forms][1] to yield the following result:

**Theorem.**
If $A = (a_{ij})$ is a positive-definite $n \times n$ matrix, then for every $x \in \mathbb{R}^n$ we have that
$$
\frac{\det(A)}{n^{n-1}a_{11} a_{22} \cdots a_{nn}} \sum a_{ii} x_i^2 \leq \sum a_{ij} x_i x_j \leq n \sum a_{ii} x_i^2 .
$$  

The inequality on the left would be greatly improved if we had the sharp upper bound required in the question. This in turn would yield a better answer to this [enclosure problem][2] (see my answer to that question).


  [1]: http://www.math.tifr.res.in/~publ/ln/tifr07.pdf
  [2]: http://mathoverflow.net/questions/87929/ellipsoids-and-lattices-an-enclosure-problem