On the Analysis of Boolean Functions and Fourier-Entropy-Influence Conjecture

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that readers could probably read it independently. Among the new results, what is the most remarkable is that we prove that the entropy of a boolean function $f$ could be upper bounded by $O(I(f))+O(\sum_{k}I_k(f)\log (1/I_k(f)))$. This is possibly the only untrivial bound for the entropy up to now