Bounds on the sum of L1 influences

Let f : {−1, 1} n → [−1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L1 influence of f can also be bounded as a function of d. Baˇckurs and Bavarian answered this question in the affirmative, providing a bound of O(d3) for general functions and O(d2) for homogeneous functions. We improve on their resultsby providing a bound of d2 for general functions and O(d log d) for homogeneous functions.