Titu's Lemma

Revision as of 01:07, 14 July 2021 by Thebeast5520 (talk | contribs)

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.