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Difference between revisions of "Nesbitt's Inequality"

 
(added another proof (original?))
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with equality when all the variables are equal.
 
with equality when all the variables are equal.
  
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Most of the proofs below generalize to proof the following stronger inequality.
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All of the proofs below generalize to proof the following stronger inequality.
  
 
If <math> a_1, \ldots a_n </math> are positive and <math> \sum_{i=1}^{n}a_i = s </math>, then  
 
If <math> a_1, \ldots a_n </math> are positive and <math> \sum_{i=1}^{n}a_i = s </math>, then  
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</center>
 
</center>
 
which follows from [[AM-HM]].
 
which follows from [[AM-HM]].
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=== By Weighted AM-HM ===
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 +
We may normalize so that <math> \displaystyle a+b+c =1 </math>.
 +
 +
We first note that by the [[rearrangement inequality]],
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<center>
 +
<math>
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3 (ab + bc + ca) \le a^2 + b^2 + c^2 + 2(ab + bc + ca)
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</math>,
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</center>
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so
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<center>
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<math>
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\frac{1}{a(b+c) + b(c+a) + c(a+b)} \ge \frac{1}{\frac{2}{3}(a+b+c)^2} = \frac{3}{2}
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</math>.
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</center>
 +
 +
Since <math> \displaystyle a+b+c = 1 </math>, weighted AM-HM gives us
 +
<center>
 +
<math>
 +
a\cdot \frac{1}{b+c} + b \cdot \frac{1}{c+a} + c \cdot \frac{1}{a+b} \ge \frac{1}{a(b+c) + b(c+a) + c(a+b)} \ge \frac{3}{2}
 +
</math>.
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</center>

Revision as of 15:31, 29 April 2007

Nesbitt's Inequality is a theorem which, although rarely cited, has many instructive proofs. It states that for positive $\displaystyle a, b, c$,

$\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge \frac{3}{2}$,

with equality when all the variables are equal.

All of the proofs below generalize to proof the following stronger inequality.

If $a_1, \ldots a_n$ are positive and $\sum_{i=1}^{n}a_i = s$, then

$\sum_{i=1}^{n}\frac{a_i}{s-a_i} \ge \frac{n}{n-1}$,

or equivalently

$\sum_{i=1}^{n}\frac{s}{s-a_i} \ge \frac{n^2}{n-1}$,

with equality when all the $\displaystyle a_i$ are equal.

Proofs

By Rearrangement

Note that $\displaystyle a,b,c$ and $\frac{1}{b+c} = \frac{1}{a+b+c -a}$, $\frac{1}{c+a} = \frac{1}{a+b+c -b}$, $\frac{1}{a+b} = \frac{1}{a+b+c -c}$ are sorted in the same order. Then by the rearrangement inequality,

$2 \left( \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \right) \ge \frac{b}{b+c} + \frac{c}{b+c} + \frac{c}{c+a} + \frac{a}{c+a} + \frac{a}{a+b} + \frac{b}{a+b} = 3$.

For equality to occur, since we changed ${} a \cdot \frac{1}{b+c} + b \cdot \frac{1}{c+a}$ to $b \cdot \frac{1}{b+c} + a \cdot \frac{1}{c+a}$, we must have $\displaystyle a=b$, so by symmetry, all the variables must be equal.

By Cauchy

By the Cauchy-Schwarz Inequality, we have

$[(b+c) + (c+a) + (a+b)]\left( \frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b} \right) \ge 9$,

or

$2\left( \frac{a+b+c}{b+c} + \frac{a+b+c}{c+a} + \frac{a+b+c}{a+b} \right) \ge 9$,

as desired. Equality occurs when $\displaystyle (b+c)^2 = (c+a)^2 = (a+b)^2$, i.e., when $\displaystyle a=b=c$.

We also present three closely related variations of this proof, which illustrate how AM-HM is related to AM-GM and Cauchy.

By AM-GM

By applying AM-GM twice, we have

$[(b+c) + (c+a) + (a+b)] \left( \frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b} \right) \ge 3 [(b+c)(c+a)(a+b)]^{\frac{1}{3}} \cdot \left(\frac{1}{(b+c)(c+a)(a+b)}\right)^{\frac{1}{3}} = 9$,

which yields the desired inequality.

By Expansion and AM-GM

We consider the equivalent inequality

$[(b+c) + (c+a) + (a+c)]\left( \frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b} \right) \ge 9$.

Setting $\displaystyle x = b+c, y= c+a, z= a+b$, we expand the left side to obtain

$3 + \frac{x}{y} + \frac{y}{x} + \frac{y}{z} + \frac{z}{y} + \frac{z}{x} + \frac{x}{z} \ge 9$,

which follows from $\frac{x}{y} + \frac{y}{x} \ge 2$, etc., by AM-GM, with equality when $\displaystyle x=y=z$.

By AM-HM

The AM-HM inequality for three variables,

$\frac{x+y+z}{3} \ge \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}$,

is equivalent to

$(x+y+z) \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \ge 9$.

Setting $\displaystyle x=b+c, y=c+a, z=a+b$ yields the desired inequality.

By Substitution

The numbers $\displaystyle x = \frac{a}{b+c}, y = \frac{b}{c+a}, z = \frac{c}{a+b}$ satisfy the condition $\displaystyle xy + yz + zx + 2xyz = 1$. Thus it is sufficient to prove that if any numbers $\displaystyle x,y,z$ satisfy $\displaystyle xy + yz + zx + 2xyz = 1$, then $x+y+z \ge \frac{3}{2}$.

Suppose, on the contrary, that $x+y+z < \frac{3}{2}$. We then have $\displaystyle xy + yz + zx \le \left( \frac{x+y+z}{3} \right)^2 < \frac{3}{4}$, and $2xyz \le 2 \left( \frac{x+y+z}{3} \right)^3 < \frac{1}{4}$. Adding these inequalities yields $\displaystyle xy + yz + zx + 2xyz < 1$, a contradiction.

By Normalization and AM-HM

We may normalize so that $\displaystyle a+b+c = 1$. It is then sufficient to prove

$\frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b} \ge \frac{9}{2}$,

which follows from AM-HM.

By Weighted AM-HM

We may normalize so that $\displaystyle a+b+c =1$.

We first note that by the rearrangement inequality,

$3 (ab + bc + ca) \le a^2 + b^2 + c^2 + 2(ab + bc + ca)$,

so

$\frac{1}{a(b+c) + b(c+a) + c(a+b)} \ge \frac{1}{\frac{2}{3}(a+b+c)^2} = \frac{3}{2}$.

Since $\displaystyle a+b+c = 1$, weighted AM-HM gives us

$a\cdot \frac{1}{b+c} + b \cdot \frac{1}{c+a} + c \cdot \frac{1}{a+b} \ge \frac{1}{a(b+c) + b(c+a) + c(a+b)} \ge \frac{3}{2}$.

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