It is well-known that if $f$ is a Boolean function of degree $d$ then its total influence is bounded by $d$. There are several ways of extending the definition of influence for non-Boolean functions. The usual way is to define the influence of the $i$th variable as the $L_2$ norm of the discrete derivative in direction $i$. Under this definition, the total influence of a bounded function (bounded by 1 in magnitude) is still upper-bounded by the degree.
Aaronson and Ambainis asked whether total $L_1$ influence can be bounded polynomially by the degree, and this was answered affirmatively by Bačkurs and Bavarian, who showed an upper bound of $O(d^3)$ for general functions, and $O(d^2)$ for homogeneous functions. We improve their results by giving an upper bound of $d^2$ in the general case and $O(d\log d)$ in the homogenous case. Our proofs are also much simpler. We also give an almost optimal bound for monotone functions, $d/2\pi + o(d)$.