On the growth of the optimal constants of the multilinear Bohnenblust-Hille inequality

Let (Kn)n=1∞ be the optimal constants satisfying the multilinear (real or complex) Bohnenblust-Hille inequality. The exact values of the constants Kn are still waiting to be discovered since eighty years ago, with the publication of Bohnenblust and Hille paper in the Annals of Mathematics. Recently, it was proved that (Kn)n=1∞ has a subpolynomial growth. Moreover it is now known that if there is an L∈[−∞,∞] such thatlimn→∞(Kn−Kn−1)=L, then L=0. In this note we show that if there is an L∈[0,∞] such thatlimn→∞K2nKn=L, thenL∈[1,D], with D=e12−12γ for real scalars and D=e1−12γ2 for complex scalars (here γ is the famous Euler-Mascheroni constant). We show that this result generalizes the former.