Difference between revisions of "Titu's Lemma"

(Created page with "Titu's lemma states that: <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...")
 
 
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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
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It is a direct consequence of Cauchy-Schwarz theorem.
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Titu's lemma is named after Titu Andreescu, and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality.

Latest revision as of 18:19, 11 October 2019

Titu's lemma states that:

\[\frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }.\]

It is a direct consequence of Cauchy-Schwarz theorem.

Titu's lemma is named after Titu Andreescu, and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality.

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