# An identity about the Majority function?

Let $f:\{-1,1\}^n \rightarrow \{-1,1\}$ be any boolean function. Let $Maj_n$ represent the majority function. Let $\langle f,g \rangle = E[f(x)g(x)]$ and $\mathcal{I}(f) = E_x[\# i, s.t. f(x)\neq f(x \oplus e_i)]$ be the total influence of a function under the standard definitions. Then does the following identity always hold?

$$\langle f,Maj_n \rangle \leq c*\frac{\mathcal{I}(f)}{\mathcal{I}(Maj_n)}$$

where c is some universal constant. I believe it might be true but have not been able to prove it. I have tried some standard functions and they seem to work. I may be mistaken as I am very new to this area. Could you please suggest a way to prove it ? or some references regarding this

• Can you prove it for a monotone function $f$? – Igor Shinkar Jan 7 '14 at 7:27