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November 3rd, 2017, 09:37 AM   #1
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Inequality proof

$\displaystyle x , y , z \in R^{+} \; $ are positive real numbers and $\displaystyle \; x+y+z=3$
prove $\displaystyle \; x^{y+z}y^{z+x}z^{x+y}\leq 1$
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November 3rd, 2017, 12:40 PM   #2
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Math Focus: Dynamical systems, analytic function theory, numerics
Here is a rough sketch for you to fill in the details.

1. Prove that $x^{y+z}y^{x+z}z^{x+y} \leq (xyz)^3$.
Hint: Use the weighted AM/GM inequality:
\[ \left( \prod_{j=1}^n a_j^{w_j} \right)^{\frac{1}{W}} \leq \frac{1}{W} \sum_{j = 1}^{n} w_ja_j \]
where $W = \sum_{j = 1}^{n} w_j$ where $a_j$ are each non-negative.

2. Let $f(x,y,z) = (xyz)^3$ and let $D = \{(x,y,z) : x + y + z =1, x,y,z \geq 0 \}$. $f$ is continuous which means $D$ is closed so $f$ attains its maximum on $D$. Note that $f = 0$ on $\partial D$ so the inequality is satisfied there.

On the interior one finds maxima of $f$ satisfying $x + y + z = 1$ only if $\nabla f$ is parallel to $(1,1,1)$. This is equivalent to $xy = yz = xz$ which is only satisfied when $x = y = z = 1$ and we see that in this case $x^{y+z}y^{x+z}z^{x+y} = 1$. Hence, the inequality holds for all $(x,y,z) \in D$ and in particular, on the interior where $x + y + z = 1$ and $x,y,z$ are positive.

Alternatively, another application of AM/GM gives you immediately that
\[(xyz)^3 \leq (\frac{x + y + z}{3})^3 = 1\]
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Last edited by SDK; November 3rd, 2017 at 12:55 PM.
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November 5th, 2017, 01:16 AM   #3
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I post quick method ,
$\displaystyle y=(3-x)lnx$
$\displaystyle y''<0 \Rightarrow y(a)+y(b)+y(c) \leq 3y(\frac{a+b+c}{3})$

$\displaystyle \ln a^{3-a} + \ln b^{3-b} +\ln c^{3-c} \leq 3y(\frac{a+b+c}{3})=1$
$\displaystyle \ln a^{3-a}b^{3-b}c^{3-c}\leq 1$
$\displaystyle a^{3-a}b^{3-b}c^{3-c}=a^{b+c}b^{c+a}c^{a+b}\leq 1$
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