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Let us consider the Arithmetic Mean -- Geometric Mean inequality for nonnegative real numbers:

$$ GM := (a_1 a_2 \ldots a_n)^{1/n} \le \frac{1}{n} \left( a_1 + a_2 + \ldots + a_n \right) =: AM. $$

It is known that the converse inequality ($\ge$) holds if and only if all the $a_i$'s are the same.

Therefore, we can expect that if the $a_i$'s are almost the same, then a converse inequality almost holds. For example, we may look for an inequality of the form $AM \le GM + f(\Delta,n)$ where $\Delta$ is the ratio between $\max_i a_i$ and $\min_i a_i$, but this is just one possibility.

Are there any natural ways to formalize the above intuition?

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I'm sure this has been done before, but you can try to work something out yourself by just setting each $a_i = a + \epsilon_i$ (or $a_i = (1 + \epsilon_i)a$ and expanding the left side up to first order in the $\epsilon_i$ and an error term that is quadratic in $\epsilon_i$ –  Deane Yang Feb 20 '13 at 17:15

7 Answers 7

up vote 12 down vote accepted

Power mean inequality can give many bounds for the difference between AM and GM. Most simple is $$AM - GM \leq \max_i a_i - \min_i a_i.$$ Another bound is $$AM - GM \leq AM - HM = \frac{a_1+\dots+a_n}{n} - \frac{n}{1/a_1 + \dots + 1/a_n}$$ etc.


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Upper and lower bounds on on the difference between the arithmetic and geometric means were given in the recent paper available at Bull. Austr. Math. Soc. or arXiv; these bounds are exact in their own terms. Also, see bibliography there.

In particular, the mentioned upper bound on $\frac1n\,\sum_1^n a_i-(a_1\cdots a_n)^{1/n}$ is $$\max\Big[\frac2n\sum_1^n(b_i-\overline b)^2,\frac1n\sum_1^n(b_i-b_\min)^2\Big], $$ where $b_i:=\sqrt{a_i}$, $\overline b:=\frac1n\,\sum_1^n b_i$, and $b_\min:=\min_{1\le i\le n} b_i$.

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I have specified the exact upper bound on the difference AM $-$ GM. –  Iosif Pinelis Sep 13 at 3:04

Here is an old result of Siegel that is related to your question.


$$ s=s(a_1,\dotsc, a_n)=\frac{1}{n} (a_1+\cdots +a_n), $$

$$ p= p(a_1,\dotsc, a_n), $$

$$ \Delta= \Delta(a_1,\dotsc, a_n)=\prod_{i,j}(a_i-a_j)^2. $$

The AM-GM inequality reads

$$\frac{s^n}{p}\geq 1. $$

Observe that $s$ is homogeneous of degree $1$, $p$ is homogeneous of degree $n$ and $\Delta$ is homogeneous of degree $n(n-1)$ in the variables $a_j$. In particular, the ratio

$$ R= \frac{p^{n-1}}{\Delta} $$

is homogeneous of degree $0$. Note that $\Delta=0$ when two of the numbers $a_j$ are equal. In particular, large $\Delta $ would mean that the numbers are "far from being equal". Equivalently, the larger $\Delta$ is, the more "dispersed" are the numbers $a_j$.

One can ask how dispersed can the numbers $a_j$ be given that $s$ and $p$ are fixed. In other words we ask to find

$$\max \Delta(a_1,\dotsc, a_n)$$

given that

$$s(a_1,\dotsc, a_n)=s_0,\;\;p(a_1,\dotsc, a_n)=p_0. $$

This constrained maximum exists and can be described explicitly as the discriminant of a certain Laguerre polynomial. I will denote it by $\Delta_\max(s_0,p_0)$.

I will set

$$ \rho=\rho(s_0,p_0)= \frac{p_0^{n-1}}{\Delta_\max(s_0,p_0)}. $$

Then there exists an explicit but very complicated strictly decreasing continuous function

$$ F_n: (0,\infty)\to (1,\infty) $$

such that

$$\lim_{t\to\infty} F_n(t)=1, $$

$$\frac{s(a_1,\dotsc,a_n)^n}{p(a_1,\dotsc,a_n)}= \frac{s_0^n}{p_0}= F_n(\rho)= F_n\left( \frac{p_0^{n-1}}{\Delta_\max(s_0,p_0)}\right) \geq F_n\left(\frac{p(a_1,\dotsc, a_n)^{n-1}}{\Delta(a_1,\dotsc, a_n)}\right). $$

Here are a few more things things about the function $F_n$. It is described as a composition $Q_n\circ P_n^{-1}$, were

$$ Q_n: (0,\infty)\to (1,\infty) $$

is a strictly decreasing, very explicit rational function and

$$P_n:(0,\infty)\to (0,\infty) $$

is a very explicit and strictly increasing polynomial such that $P_n(0)=0$. This implies the sharper inequality

$$ s(a_1, \dotsc, a_n)^n \geq F_n\left(\frac{p(a_1,\dotsc, a_n)^{n-1}}{\Delta(a_1,\dotsc, a_n)}\right)p(a_1,\dotsc, a_n), $$

with equality iff

$$ \Delta(a_1,\dotsc,a_n)=\Delta_\max(s,p). $$

For more details see Sec. 8.6 of the beautiful book Special Functions by G.E. Andrews, R. Askey, R. Roy.

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A result due to Cartwright and Field gives an upper bound for $AM-GM$ of the form you seek:

$$AM-GM \le \frac{1}{2n \min a_i} \sum_{i=1}^{n} (a_i-AM)^2$$ A very naive computation shows that the RHS is $\le \frac{1}{2 \min a_i} (\max a_i - \min a_i)^2$.

This estimate is already better than the suggested naive bound $\max a_i - \min a_i$ when $\max a_i$ and $\min a_i$ are relatively close, specifically: when $\max a_i \le 3 \min a_i$.

It is also worth mentioning that this upper bound was further improved by several authors.

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Proposition 1 in this paper might be what you are looking for.

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The left and right sides are both continuous functions.

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It's not precisely what you asked about, but this paper by Gluskin and Milman shows that, for "most" sequences $a_1, \dotsc, a_n$, the AM-GM inequality can be reversed up to a multiplicative constant. The paper contains a number of observations which come closer to directly addressing your question.

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Thanks, it's a very interesting answer even though, as you observed, not exactly what I was looking for. –  Vincenzo Feb 21 '13 at 13:15

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