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

Abstract

Simple proofs are given of improved results of Brown and Shepp which are useful in calculations with fractal sets. A new inequality for convex functions is also obtained.

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.…”

“…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. This paper is a continuation of [17], from which we take the terminology and notation. We are able to improve the main theorems of [17], due to improvements in the auxiliary results and lemmas.…”

“…This paper is a continuation of [17], from which we take the terminology and notation. We are able to improve the main theorems of [17], due to improvements in the auxiliary results and lemmas. In Section 2 we give a strengthened form of the main lemma underlying the results of [1].…”

“…The improvements to the basic lemma given in the previous section have direct consequences for several theorems given in [17]. The proofs of the first two involve only straightforward modifications of those used in earlier improvements presented in [1] and so are omitted.…”

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AbstractWe improve some results of [17], which relate to key tools given in [7] for establishing canonical inequalities used in the analysis of sum sets and fractals.

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