Titu's Lemma

Titu's lemma states that:

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.