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.
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It is a direct consequence of Cauchy-Schwarz theorem as follows,
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\begin{align*}
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\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,\
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\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 }
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\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 01:07, 14 July 2021

Titu's lemma states that:

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

It is a direct consequence of Cauchy-Schwarz theorem as follows, (a12b1+a22b2++an2bn)(b1+b2++bn)(a1+a2++an)2,a12b1+a22b2++an2bn(a1+a2++an)2b1+b2++bn

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