On the maximal L1 influence of real-valued boolean functions

We show that any sequence of well-behaved (e.g. bounded and non-constant) real-valued functions of $n$ boolean variables $\{f_n\}$ admits a sequence of coordinates whose $L^1$ influence under the $p$-biased distribution, for any $p\in (0,1)$, is $\mathrm{\Omega }(\text{var}(f_n)\frac{\ln n}{n})$.