Comments on some inequalities of Pearce and Pečarić

Abstract

We prove inequalities for convex functions, V norms, and sums of powers. Our results sharpen recently published inequalities of C. E. M. Pearce and J. E. Pecaric.1991 Mathematics subject classification: 26D15.

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.…”

“…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]. Improved forms of several existing theorems follow immediately and are given in Section 3.…”

“…- Also, the remark in [1] that the constant factor p/(n -1 ) cannot be replaced by a greater number is not completely correct. As we see, this factor can be replaced by p'/(n -1), which may be greater if p v -p.…”

“…The proofs of the first two involve only straightforward modifications of those used in earlier improvements presented in [1] and so are omitted. The following result is an improvement of Theorem 1 from [17].…”