Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures

Let mu be a Borel probability measure on R-d. We study the Hausdorff dimension and the packing dimension of the multifractal Hausdorff measure H-mu(q,t) and the multifractal packing measure P-mu(q,t) introduced in [L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82-196]. Let b(mu) denote the multifractal Hausdorff dimension function and let B-mu denote the multifractal packing dimension function introduced in [Olsen, op cit]. For a fixed q is an element of R, we obtain bounds for the Hausdorff dimension and the packing dimension of H-mu(q,b mu(q)) and P-mu(q,B mu(q)) in terms of the subdifferential of b(mu) and B-mu at g. For q = 1, our result reduces to [GRAPHICS] where D-Bmu(1) and D+Bmu(1) denote the left and right derivative of B-mu at 1. Inequality (*) improves a similar result obtained independently by Y. Heurteaux and S.-Z. Ngai. It follows from (*) that if the mulifractal box dimension spectrum (or L-q spectrum) tau(mu) of mu is differentiable at 1 then -tau(mu)'(1) equals the entropy dimension (or information dimension) of mu. This result has been conjectured in the physics literature and proved rigorously in certain special cases.