[Soln] Only AM-GM

Using AM-GM (Arithmetic Mean-Geometric Mean Inequality), prove that for x, y, z \geq 0,

x^5yz + xy^5z + xyz^5 \geq x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3.

The inequality is not a difficult one (in fact, it is an obvious corollary of Murihead’s inequality), but I just wanted to illustrate a technique of using AM-GM.

The AM-GM can’t be directly applied in the following standard ways because the terms on the RHS don’t work out:

x^5yz + xy^5z \geq 2 \sqrt{x^6y^6z^2} = 2x^3y^3z,
x^5yz + xy^5z + zyz^5 \geq \sqrt[3]{x^7y^7z^7} = (xyz)^{7/3}.

The trick here is that terms can be repeated multiple times in the AM-GM inequality. For example, instead of directly applying the AM-GM like this:

3x + y \geq 2\sqrt{3xy},

We could do this to get a different lower bound:

3x + y = x + x + x + y \geq 4 \sqrt[4]{x^3y}.

Applying this to our problem: we need to repeat the terms on the LHS an appropriate number of times such that the exponents of x,y,z work out to be the exponents on the RHS. If we let the 3 terms on the LHS be repeated a,b,c times respectively, we have

ax^5yz + bxy^5z + cxyz^5 \geq (a+b+c) \sqrt[a+b+c]{x^{5a + b+ c}y^{a+5b+c}z^{a+b+5c}}.

To make the RHS of the inequality above match the terms in the RHS of the original inequality, we need:

\begin{cases} 5a + b + c &= 3(a+b+c), \\  a + 5b + c &= 2(a+b+c), \\  a + b + 5c &= 2(a+b+c). \end{cases}

These can be solved through the usual method of elimination to give the solution (a,b,c) = (2, 1, 1). (Note: The solution could have been guessed by some trial and error, but the above is a general method that doesn’t require any guessing.) Hence,

x^5yz + x^5yz + xy^5z + xyz^5 \geq 4x^3y^2z^2.

Similarly we find that

\begin{aligned} x^5yz + xy^5z + xy^5z + xyz^5 &\geq 4x^2y^3z^2, \\  x^5yz + xy^5z + xyz^5 + xyz^5 &\geq 4x^2y^2z^3. \end{aligned}

Adding up the last 3 inequalities and dividing by 4, we obtain the given inequality. Done!

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One Response to [Soln] Only AM-GM

  1. Pingback: [Soln] 2015 Romania Junior Balkan TST Day 1 Problem 3 | Beyond Solutions

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