Read-once formulas and a question of Han on the Fourier entropy-influence conjecture

For a Boolean function f and a coordinate k, let Qₖ(f)=∑S k| f(S) f(S∪\)|/Iₖ(f), where Iₖ(f) is the influence of k. In his proof of a new bound towards the Fourier entropy-influence conjecture, Han asked whether there is a universal constant c such that every Boolean function has a coordinate k with Qₖ(f)≤ c, and observed that this would imply the conjecture. We show that the answer is negative. For balanced alternating read-once AND-OR formulas we compute Qₖ exactly by a one-step recursion over the levels of the formula; it is the same at every coordinate. Choosing the fan-ins so that every gate below the root is true with probability at most $1/4$ gives monotone read-once formulas of depth L on at most 2· 7L-1 variables with Qₖ(f)≥ L for every k, so minₖ Qₖ(f) grows at least like log₂ n/log₂ 7. Depth two gives the tribes functions, for which minₖ Qₖ approaches $4$ but stays below it. The Fourier entropy-influence conjecture is known for these formulas, so the inequality in Han's question fails on a class where the conjecture holds. By contrast, the drop of the Fourier entropy under splitting on a coordinate is less than 5+ln 4 times the influence of that coordinate, for every read-once formula and every coordinate.