  This note is concerned with an extension, at second order, of an inequality
on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to
Talagrand (\cite{TalL1L2}). As an application, the main result of this note is
a Theorem in the spirit of a famous result from Kahn, Kalai and Linial (cf.
\cite{KKL}) concerning the influence of Boolean functions. The notion of the
influence of a couple of coordinates $(i,j)\in\{1,\ldots,n\}^2$ is introduced
in section 2 and the following alternative is obtained : for any Boolean
function $f\,:\, C_n\to \{0,1\}$, either there exists a coordinate with
influence at least of order $(1/n)^{1/(1+\eta)}$, with $\, 0<\eta<1$
(independent of $f$ and $n$) or there exists a couple of coordinates
$(i,j)\in\{1,\ldots,n\}^2$ with $i\neq j$, with influence at least of order
$(\log n/n)^2$. In section 4, it is shown that this extension of Talagrand
inequality can also be obtained, with minor modifications, for the standard
Gaussian measure $\gamma_n$ on $\mathbb{R}^n$ ; the obtained inequality can be
of independent interest. The arguments rely on interpolation methods by
semigroup together with hypercontractive estimates. At the end of the article,
some related open questions are presented.
