A new bound for the Fourier-Entropy-Influence conjecture

In this paper, we prove that the Fourier entropy of an $n$-dimensional boolean function $f$ can be upper-bounded by $O(I(f)+ \sum\limits_{k\in[n]}I_k(f)\log \frac{1}{I_k(f)})$, where $I(f)$ is its total influence and $I_k(f)$ is the influence of the $k$-th coordinate. The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels.