On two lemmas of Brown and Shepp having application to sum sets and fractals

Abstract

An improvement is made to two results of Brown and Shepp which are useful in calculations with fractal sets.

On two lemmas of Brown and Shepp having application to sum sets and fractals, II

“…In [1], Brown and Shepp provided two key lemmas which have proved valuable in making available a number of simple calculations in this area. Improvements of the results of Brown and Shepp were obtained in [2]. Further generalizations of these results are given in [3].…”

“…Recently there has been a resurgence of interest in sum sets, which have, inter alia, application to fractals, iterated function systems and dynamical systems (see the authors [2] for some select references in the area). The calculation of associated Hausdorff dimensions and Hausdorff measures and other properties can be delicate.…”

On two lemmas of Brown and Shepp having application to sum sets and fractals, III

“…Establishing the canonical univariate inequalities can be quite tricky and some effort has been put into sharpening techniques for their derivation (see [7,11,[14][15][16][17] and most recently [1]). The 'two lemmas' of the title are special cases of the two parts of Theorem 3.2 below, the earliest versions of which are due to Brown and Shepp [7] and have influenced further work in the area.…”

Comments on some inequalities of Pearce and Pečarić

“…Theorem B. (Pearce and Pecaric,[6]). Let the positive real numbers a, b, s, and t, (/ = 0, 1, 2) satisfy a/s, + b/t, = 1 (i = 0, 1, 2) and s, < s Q < s 2 .…”