  The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96]
seeks to relate two fundamental measures of Boolean function complexity: it
states that $H[f] \leq C Inf[f]$ holds for every Boolean function $f$, where
$H[f]$ denotes the spectral entropy of $f$, $Inf[f]$ is its total influence,
and $C > 0$ is a universal constant. Despite significant interest in the
conjecture it has only been shown to hold for a few classes of Boolean
functions.
  Our main result is a composition theorem for the FEI conjecture. We show that
if $g_1,...,g_k$ are functions over disjoint sets of variables satisfying the
conjecture, and if the Fourier transform of $F$ taken with respect to the
product distribution with biases $E[g_1],...,E[g_k]$ satisfies the conjecture,
then their composition $F(g_1(x^1),...,g_k(x^k))$ satisfies the conjecture. As
an application we show that the FEI conjecture holds for read-once formulas
over arbitrary gates of bounded arity, extending a recent result [OWZ11] which
proved it for read-once decision trees. Our techniques also yield an explicit
function with the largest known ratio of $C \geq 6.278$ between $H[f]$ and
$Inf[f]$, improving on the previous lower bound of 4.615.
