This article fits in many studies of multifractal analysis of measure [1, 2, 3, 4, 6, 7, 8, 9]. We took as a starting point the work of F. Ben Nasr in [2] to give a new inequality involving Dim(X α ) which would be, in certain cases, finer than the inequality Dim(X α ) ≤ inf q≥0 (αq +Bμ(q)), established by L. Olsen in [6]. Besides we elaborated an application of our result which gives a better ...