# new problem (vankhea)

Let $m, M, a_i, b_i, \alpha, \beta, \gamma$ be positive real numbers such that:
$\displaystyle m\leq \frac{a_i}{b_i}\leq M$ and $\alpha -\beta +\gamma=1, \forall{\beta =min(\alpha, \beta, \gamma)}$.
Prove that: $\displaystyle (\alpha m+\gamma M)\sum_{i=1}^{n}a_i^{\beta}b_i\geq m^{\alpha}M^{\gamma}\sum_{i=1}^{n}b_i^{\beta +1}+\beta \sum_{i=1}^{n}a_i^{\beta+1}$