Journal of Inequalities and Applications
Volume 2 (1998), Issue 3, Pages 229-233

On an inequality conjectured by T.J. Lyons

E. R. Love

Department of Mathematics, The University of Melbourne, Parkville 3052, Victoria, Australia

Received 1 April 1997

Copyright © 1998 E. R. Love. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let n be any positive integer, x and y any positive real numbers. The inequality αj=0n(αn)!(αj)!(α(nj))!xαjyα(nj)(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,