Abstract. The inequality $C_2(n)\leq2 K^\C_G$, where $K_G^\C $ is the complex Grothendieck constant and $C_2(n)=\sup\big \{\|p(\boldsymbol T)\|:\|p\|_{\D ^n,\infty }\leq1, \|\boldsymbol T\|_{\infty } \leq1 \big\},$ for each $n\in\N,$ is due to Varopoulos. Here the supremum is taken over all commuting $n$-tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions and all complex polynomials $p$ in $n$ variables of degree at most $2$ and of supremum norm at most $1$ over the polydisc. We show that $C_2(n)\leq\frac {3\sqrt{3}}{4} K^\C_G$ for each $n\in\N.$

DOI: 10.14232/actasm-015-088-4

AMS Subject Classification (1991): 47A13

Keyword(s): complex Grothendieck constant, von Neumann inequality, Varopoulos-Kaijser polynomial

Received December 18, 2015, and in final form February 17, 2016. (Registered under 88/2015.)