Polynomial and multilinear Hardy-Littlewood inequalities: analytical and numerical approaches

Abstract

Abstract. We investigate the constants of the polynomial and multilinear Hardy-Littlewood inequalities. Among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araújo et al. In a final section, as an independent appendix, we present some computer-aided estimates for the lower bounds of the multilinear Hardy-Littlewood inequalities.Mathematics subject classification (2010): 46G25, 47L22, 47H60.

New Lower Bounds for the Constants in the Real Polynomial Hardy–Littlewood Inequality

“…The estimates obtained in this section can be summarized as follows: Proof. The parameters that we found for a, b, c, d are the following, which improve the estimates of the previous section we note that the estimates are (we note that when trying to find the coefficients we have numerical evidence that the best constant for the case m = 2 is given by the polynomial of the polynomial P 2 of the previous section: The following In [9,15] it is defined: For the real case, it has been recently proved in [10] that H R,p ≥ 2. However, not many exact values of H R,p (n) are known so far.…”

Towards sharp Bohnenblust–Hille constants

“…Since λ m−1 = s, applying the case i = k we can estimate We can now establish the following lower bound estimate for the entropy in the Hardy-Littlewood inequality. First, we recall a result from [9], which will be used herein as technical lemma and we sketch its proof for the sake of completeness:…”