The multilinear Bohnenblust-Hille inequality (for real scalars) states that for all \(m\), there exists a constant \(C_{m}\geq 0\) such that, for every positive integer \(N\) and every \(m\)-linear mapping \(T:\ell _{\infty }^{N}\times \dots \times \ell _{\infty }^{N}\rightarrow \mathbb{R}\), \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{N}\left| T\left( e_{i_{1}},\dots,e_{i_{m}}\right) \right| ^{\frac{2m}{m+1}}\right) ^{\frac{ m+1}{2m}}\leq C_{m}\left| T\right| . \] The case \(m=2\) is the well-known Littlewood \(4/3\) inequality [J.E.Littlewood, Q.J.Math., Oxf.Ser.1, 164–174 (1930; JFM 56.0335.01)].
Since its publication in 1931, several authors have obtained upper estimates for the values of \(C_{m}\). It is clear that \(C_{m}\geq 1\), but no better lower bound seemed to be known for general \(m\) until now.
In this paper, the aim of the authors is to give a nontrivial lower bound for \(C_{m}\). The tool is a clever and nice induction argument, in which the definition \[ \max \{a,b\}:=\frac{\left| a+b\right| +\left| a-b\right| }{2} \] is advantageously applied, in order to give the lower bound for the Bohnenblust-Hille constant \[ C_{m}\geq 2^{\frac{m-1}{m}}. \] This estimate has an extra bonus, because in the case \(m = 2\) the optimal constant \(C_2\) in the Littlewood \(4/3\) inequality is obtained.