Abstract
Let n be any positive integer, x and y any positive real numbers. The inequality
α∑j=0n(αn)!(αj)!(α(n−j))!xαjyα(n−j)≤(x+y)αn
was conjectured for 0<α<1 by T.J. Lyons, after he had proved it with an extra factor 1/α on the right, in a preprint (Imperial College of Science, Technology and Medicine, 1995). Many numerical trials confirmed the conjecture, and none disproved it. The present paper proves it, with strict inequality, for all a in sufficiently small neighbourhoods of 12,14,18,⋯