On the Bohnenblust–Hille inequality and a variant of Littlewoodʼs 4/3 inequality

The search for sharp constants for inequalities of the type Littlewoodʼs 4/3 and Bohnenblust–Hille has lately shown unexpected applications in many fields such as Analytic Number Theory, Quantum Information Theory, or in results on n-dimensional Bohr radii. Recent estimates obtained for the multilinear Bohnenblust–Hille inequality (for real scalars) have been used, as a crucial tool, by A. Montanaro in order to solve problems in Quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust–Hille constants (for complex scalars). For bilinear forms, we provide optimal constants of variants of Littlewoodʼs 4/3 inequality (for real scalars) when the exponent 4/3 is replaced by any . We also prove that the optimal constants in real case are always strictly greater than those from the complex case.