I have read may texts saying that if it's a homogeneous inequality
Then you can assume abc =1 or a+b+c =1
But what is a homogeneous inequality
Can anybody explain

A homogeneous inequality is an inequality in which all terms have the same degree. For example, \(abc^2 + a^4 + ab^3 \geq 13bc^3\) is a homogeneous inequality, since all the terms have degree 4, whereas \(a^3+ab+b^4 \geq 12\) is a heterogeneous inequality since all the terms don't have the same degree.

The term \(abc^2\) has degree 4, doesn't it? The term \(a^4\) also has degree 4. The term \(bc^3\) has degree 4, and finally, \(bc^3\) has degree 4. Therefore, the overall expression has degree 4. Recap on your polynomial definitions (including degrees of multivariate polynomials) to understand. :)

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TopNewestA homogeneous inequality is an inequality in which all terms have the same degree. For example, \(abc^2 + a^4 + ab^3 \geq 13bc^3\) is a homogeneous inequality, since all the terms have degree 4, whereas \(a^3+ab+b^4 \geq 12\) is a heterogeneous inequality since all the terms don't have the same degree.

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I didn't understand the first example Degree 4 ??

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The term \(abc^2\) has degree 4, doesn't it? The term \(a^4\) also has degree 4. The term \(bc^3\) has degree 4, and finally, \(bc^3\) has degree 4. Therefore, the overall expression has degree 4. Recap on your polynomial definitions (including degrees of multivariate polynomials) to understand. :)

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