Journal of Inequalities and Applications
Volume 3 (1999), Issue 3, Pages 285-291
doi:10.1155/S1025583499000193
A Maximal inequality of non-negative submartingale
Department of Mathematics, Chang-Won National University, Chang-Won 641-773, Korea
Received 26 March 1998; Revised 17 August 1998
Copyright © 1999 Young-Ho Kim. 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.
Abstract
In this paper, we prove the maximal inequality λP(supn≥0(fn+|gn|≥λ)≤(Q(1)+2)||f||1, λ>0, between a non-negative submartingale f, g is strongly subordinate to f and 1−2fn−1−Q(1)≤0, where Q is real valued function such that 0<Q(s)≤s for each s>0, Q(0)=0. This inequality improves Burkholder’s inequality in which Q(1)=1.