A conjecture of Blakley and Dixon about odd powers of positive matrices

In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both odd,

$$\langle u, S^ku\rangle^j\ge \langle u, S^ju\rangle^k.$$

They verify this conjecture under the condition that RHS is sufficiently large. I am wondering if there has been any progress towards this conjecture.

A year earlier, Blakley and Roy verify the $j=1$ case. This also implies that the conjecture is true whenever $j$ divides $k$. There are no papers known to Google scholar citing this paper that solves the conjecture.

• If $S$ is in addition positive definite, then clearly the conjecture is true. Otherwise, you could try to generalize the proof of T. Pate, and see if holds: ams.org/tran/2012-364-08/S0002-9947-2012-05501-2 (you may also enjoy chasing the references to Sidorenko's conjecture which is much more general than the Blakley-Roy inequality) – Suvrit Apr 16 '16 at 21:28
• Thank you, your references are very accurate, as always! I was just playing with graphon generalizations of this statement, and indeed one gets the k-path vs j-path version of Sidirenko's conjecture. – Mert Sağlam Apr 16 '16 at 21:40
• When S is PSD or k,j are even, then indeed the series $s_k = \langle u, S^k u\rangle$ is completely monotone, by Hausdorff moment problem. This implies for instance the log-convexity and the claim of the conjecture. (My interest/ attempt in the moment problem on [-1,1] that I posted here earlier this week is coming from this application) – Mert Sağlam Apr 16 '16 at 21:45
• Indeed, Pate's paper seems to be a great resource. – Mert Sağlam Apr 16 '16 at 21:46
• I'd be quite curious to hear if you discover any extension to Pate's ideas! – Suvrit Apr 16 '16 at 21:47