On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality

TL;DR

This paper investigates the Bohnenblust-Hille and Littlewood's 4/3 inequalities, providing new bounds and optimal constants, with implications for various mathematical and applied fields such as quantum information theory.

Contribution

It introduces new upper bounds for the Bohnenblust-Hille constants over complex scalars and determines optimal constants for variants of Littlewood's 4/3 inequality for real scalars.

Findings

New upper bounds for Bohnenblust-Hille constants in complex scalars

Optimal constants for variants of Littlewood's 4/3 inequality in real scalars

Real case constants are strictly greater than complex case constants

Abstract

The search for sharp constants for inequalities of the type Littlewood's 4/3 and Bohnenblust-Hille, besides its pure mathematical interest, has shown unexpected applications in many different fields, such as Analytic Number Theory, Quantum Information Theory, or (for instance) in deep results on the $n$-dimensional Bohr radius. The recent estimates obtained for the multilinear Bohnenblust-Hille inequality (in the case of real scalars) have been recently used, as a crucial step, by A. Montanaro in order to solve problems in the theory of quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust-Hille constants in the case of complex scalars. For bilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3 inequality (in the case of real scalars) when the exponent 4/3 is replaced by any $r\geq4/3.$ As a consequence of our estimates we show that the optimal constants for the real case are always strictly greater than the constants for the complex case.