This is a followup to this question Is this AM/GM refinement correct or not?. In it, as joriki helpfully pointed out, the correct form of a generalization of AM/GM proved by Dragomir is as follows. Let $p_i=1$ and $x_i\in[a,b]$ then the following holds: $ \exp\left(\frac{1}{2b^2n^2}\sum\limits_{i
Now I am asking the following question. What is an effective lower bound on the $A_n/G_n$ ratio for $x_i\in[0,1]$ if we know that $\min|x_i-x_j|\ge p$, i.e. the numbers are not close to each other.
Case n=2
Let $f_2(x,y)=(x+y)/(2 \sqrt{x y})$ and assume $|x-y|\ge p$. Using Dragomir's inequality we get the following lower bound: $f(x,y)>\exp(p^2/8)$.
If we however examine the function $f_2(x,x-p)$ on the interval $[0,1]$ we see that it is decreasing in $x$ (and increasing in $p$) and it's minimum (for fixed $p$) is attained at $x=1$. Hence we arrive at the following lower bound $f_2(x,y)\ge f_2[1,1-p]$, or in other words: $\frac{x+y}{2 \sqrt{x y}}\ge \frac{2-p}{2 \sqrt{1-p}}>\exp(p^2/8)$
So this is an improvement on the general inequality.
Case n=3
Let $f_3(x,y,z)=(x+y+z)/\sqrt[3]{xyz}$ where $x$,$y$, and $z$ lie in [0,1] and the minimal distance between any two of them is $p<1/2$. According to the general inequality, we have $f_3(x,y,z)>\exp(p^2/3)$.
Let's examine the function $f_3(x,x-p,x-q)$ where $x\in[0,1]$ and $x>\max(p,q)$. It's derivative is surprisingly simple:$\frac{df_3(x,x-p,x-q)}{dx}=\frac{x \left(-2 p^2+2 p q-2 q^2\right)+p^2 q+p q^2}{9 x (x-p)(x-q) \sqrt[3]{x (p-x) (q-x)}}$
Simple calculations show that the minimum is achieved when $x=1$ and this minimum is: $f_3(1,1-p,1-q) = \frac{3-p-q}{3 \sqrt[3]{(1-p) (1-q)}} $ Taking again the derivative it is obvious that given a fixed $q$, the minimum is attained when $p=q/2$, or in other words $f_3(1,1-p,1-q)\ge f_3(1,1-p,1-2p)$ for $q>p$.
We have thus proved that: $\frac{x+y+z}{3 \sqrt[3]{x y z}}\ge f_3(1,1-p,1-2p)=\frac{1-p}{\sqrt[3]{2 p^2-3 p+1}}$
So again we have a better bound then the one offered by Dragomir's inequality, though I can't prove that it is better.
The questions.
- How can one prove that $\frac{1-p}{\sqrt[3]{2 p^2-3 p+1}}>\exp(p^2/3)$? I have verified this only numerically.
- How can those results be generalized for $n>3$ and hence improve the bound $A_n/G_n\ge \exp(\frac{1}{24} \left(n^2-1\right) p^2)$ which follows from Dragomir's inequality?
- Are those inequalities known already?
Hypothesis
The two cases I discussed can be straight-forward generalized to the following hypothesis for the general case. Let $q\ge n-1$, $x_i\in[0,1]$, and $|x_i-x_j|>=1/q$. Then the following inequality holds $\frac{A_n(x)}{G_n(x)}>\left(\frac{1-n}{2}+q\right) \left((-1)^n \left(-q\right)_n\right)^{-1/n}$where $\left(-q\right)_n$ is the Pochhammer symbol. Note that for $n<4$ this is equivalent to what's proved above.