A new upper bound for the complex Grothendieck constant

Abstract

Let ϕ denote the real function $$\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1$$ and letK G C be the complex Grothendieck constant. It is proved thatK G C ≦8/π(k 0+1), wherek 0 is the (unique) solution to the equationϕ(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k 0+1) ≈ 1.40491. The previously known upper bound isK G C ≦e 1−y ≈ 1.52621 obtained by Pisier in 1976.