Journal of Inequalities and Applications 
Volume 3 (1999), Issue 3, Pages 285-291
doi:10.1155/S1025583499000193

A Maximal inequality of non-negative submartingale

Young-Ho Kim

 Department of Mathematics, Chang-Won National University, Chang-Won 641-773, Korea
Received 26 March 1998; Revised 17 August 1998

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.