# A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis

I have a question. The question is to prove:

The weight $w \in A_\infty$if and only if $\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all cubes $Q\in \mathbb{R}^n$, where $C$ is a constant independent of $Q$ (and $w$?).

All other equivalent conditions of $A_\infty$ can be used. I really don't know how to get the point!

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You capitalised "Real and Harmonic analysis": do you just intend the subject or are you thinking about a particular book? –  Willie Wong Jun 25 '13 at 11:11
I can't post this as a comment right now, so I am posting it as an answer, although it is certainly not nearly a full answer to your question. Notice that $\frac{1}{|Q|}\int_Q f(x)dx$ corresponds to the measure-theoretic arithmetic mean, while $\exp (\frac{1}{|Q|}\int_Q \log f(x) dx)$ corresponds to the measure-theoretic geometric mean. That would be a start. (Alternatively, Jensen's inequality can be used since you are looking at normalized integrals and $\exp$ is convex.) In other words, you are pretty much looking at the inequality between harmonic and geometric mean in measure-theoretic te –  Marin Jun 25 '13 at 12:57
That is a subject,because i thought the question is on real or harmonic analysis –  Reigion Ho Jun 25 '13 at 15:31
Thank Marin , I got the point ,and knew how to deal with it.Thanks very much. –  Reigion Ho Jun 26 '13 at 16:28