I am trying to find the upperbound of the following form,
$\frac{{{{\left( {\sum\limits_{i = 1}^M {{\varepsilon _i}{a_i}} } \right)}^2}}}{{\left( {\sum\limits_{i = 1}^M {{\varepsilon _i}{b_i}} } \right)}} \le \sum\limits_{i = 1}^M {\left( {.......} \right)} $
Where, $\sum\limits_{i = 1}^M {{\varepsilon _i}} = 1;0 \le \sum\limits_{i = 1}^M {{\varepsilon _i}{a_i}} ,\sum\limits_{i = 1}^M {{\varepsilon _i}{b_i}} \le 1;{a_i} \in R;{b_i} \in R$.
I found a lemma known as Titu Lemma (As a special Case in the link) (Titu Andreescu) which looks almost similar but $\epsilon_i$ is not present,
$\frac{{{{\left( {\sum\limits_{i = 1}^M {{a_i}} } \right)}^2}}}{{\left( {\sum\limits_{i = 1}^M {{b_i}} } \right)}} \le \sum\limits_{i = 1}^M {\frac{{{a_i}^2}}{{{b_i}}}} $
Any hints or theorem will be helpful. Thank you.