Fang and I found enhanced inequalities about the arithmetic mean and the geomertic mean of n positive numbers (see this arXiv entry for details).

Ofcourse it is impossible to improve those classic inequalities without new information. Let me explain our ideas.

Suppose we know the arithmetic mean, An, and some elements, and now we want to estimate their geometric mean, Gn. We can construct the following inequality.

Inserting the best choice, we obtain the first inequality. The second one can be obtained in a similar way.

It is no big deal to obtain these enhanced inequalities. But it is interesting to find that they are actually better than previous results (S. H. Tung, Mathematics of computation, Vol. 29, No. 131, pages 834-836, 1975).

It is a good exercise. Here is a little story. I submitted it to arXiv, but it was put on hold for two weeks. I guess if it is because the arXiv moderators think the results are too naive. Fortunately, it turns out that I put it into the wrong math category. I put it in math.NT in the beginning. But the moderator put it in math.GM in the end. It is not surprising. There are too many false proofs (minke, you know) in math.GM. For example, someone is trying to prove Fermat's last theorem using an elementry method. Even I, a non-math physician, can easily find one of his mistakes on the first page.