A classical inequality due to Bohnenblust and Hille states that for every positive integer m there is a constant Cm>0 so that (i1,...,im=1∑N∣U(ei1,...,eim)∣m+12m)2mm+1≤Cm∣U∣
Nearby in the stack
for every positive integer
N
and every
m
-linear mapping
U:ℓ∞N×...×ℓ∞N→C
, where
Cm=m2mm+122m−1.
The value of
Cm
was improved to
Cm=22m−1
by S. Kaijser and more recently H. Quéffelec and A. Defant and P. Sevilla-Peris remarked that
Cm=(π2)m−1
also works. The Bohnenblust--Hille inequality also holds for real Banach spaces with the constants
Cm=22m−1
. In this note we show that a recent new proof of the Bohnenblust--Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for
Cm
for all values of
m∈N
. In particular, we will also show that, for real scalars, if
m
is even with
2≤m≤24
, then
CR,m=21/2CR,m/2.
We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants.