Difference between revisions of "Homogeneous"

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A function <math>f(a_1,a_2,\ldots,a_n)</math> is said to be '''homogenous''' if all its terms are of the same degree in <math>a_i</math>.  
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A [[polynomial]] <math>f(a_1,a_2,\ldots,a_n)</math> is said to be '''homogeneous''' if all its terms are of the same degree in <math>a_i</math>.  
  
This concept is often used in [[inequalities]] so that one can "scale" the terms, and assume things like the sum of the involved variables is <math>1</math>.
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This concept of homogeneity is often used in [[inequalities]] so that one can "scale" the terms (this is possible because <math>f(ta_1,ta_2,\ldots,ta_n)=t^kf(a_1,a_2,\ldots,a_n)</math> for some fixed <math>k</math>), and assume things like the sum of the involved variables is <math>1</math>, for things like [[Jensen's Inequality]]
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== Problems ==
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=== Introductory ===
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=== Intermediate ===
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=== Olympiad ===
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*Let <math>a,b,c</math> be positive real numbers. Prove that
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<math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]])
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== See Also ==
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* [[Inequalities]]
  
 
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[[Category:Algebra]]
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[[Category:Inequalities]]
 
[[Category:Definition]]
 
[[Category:Definition]]
[[Category:Elementary algebra]].
 

Latest revision as of 12:38, 14 July 2021

A polynomial $f(a_1,a_2,\ldots,a_n)$ is said to be homogeneous if all its terms are of the same degree in $a_i$.

This concept of homogeneity is often used in inequalities so that one can "scale" the terms (this is possible because $f(ta_1,ta_2,\ldots,ta_n)=t^kf(a_1,a_2,\ldots,a_n)$ for some fixed $k$), and assume things like the sum of the involved variables is $1$, for things like Jensen's Inequality

Problems

Introductory

Intermediate

Olympiad

  • Let $a,b,c$ be positive real numbers. Prove that

$\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (Source)

See Also

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