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. :)

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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.

Log in to reply

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

Log in to reply

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. :)

Log in to reply

Log in to reply