Difference between revisions of "Titu's Lemma"
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<cmath> \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }. </cmath> | <cmath> \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }. </cmath> | ||
| − | It is a direct consequence of Cauchy-Schwarz theorem | + | It is a direct consequence of Cauchy-Schwarz theorem as follows, |
| + | \begin{align*} | ||
| + | \left(\frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \right) \left( b_1 + b_2 + \cdots+ b_n \right) &\geq (a_1 + a_2 + \cdots+ a_n ) ^2,\\ | ||
| + | \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } &\geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n } | ||
| + | \end{align*} | ||
Titu's lemma is named after Titu Andreescu, and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality. | Titu's lemma is named after Titu Andreescu, and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality. | ||
Revision as of 02:07, 14 July 2021
Titu's lemma states that:
![\[\frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }.\]](http://latex.artofproblemsolving.com/2/6/4/264fff665becc96f5f9e05a178f2a31ee3f2603f.png)
It is a direct consequence of Cauchy-Schwarz theorem as follows, \begin{align*} \left(\frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \right) \left( b_1 + b_2 + \cdots+ b_n \right) &\geq (a_1 + a_2 + \cdots+ a_n ) ^2,\\ \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } &\geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n } \end{align*}
Titu's lemma is named after Titu Andreescu, and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality.
