Multifractal Formalism and Inequality Involving Packing Dimension

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 inequality involving Dim(X α ). We are thankful to Mr F. Ben Nasr for the long and lucrative discussions which we had during the development of this work. 1. Multifractal formalism Let μ be a Borel probability measure on R. For E ⊂ R, q, t ∈ R and ε > 0, by adopting the convention { 0 = +∞, q < 0, 0 = 1, put P q,t μ,ε(E) = sup { ∑ i μ (B (xi, ri)) q (2ri) t } where the supremum is taken over all the centered ε−packing (B (xi, ri))i∈I of E. Also put P q,t μ (E) = lim ε→0 P q,t μ,ε(E). Since P q,t μ is a prepacking-measure, then we consider, P q,t μ (E) = inf E⊂ „