A new upper bound for the complex Grothendieck constant

Let ? denote the real function $$varphi (k) = ksmallint _0^{pi /2} frac{{cos^2 t}}{{sqrt {1 - k^2 sin ^2 t} }}dt, - 1 leqq k leqq 1$$ 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 equation?(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.