## 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: $\displaystyle (\alpha m+\gamma M)\sum_{i=1}^{n}a_i^{\beta}b_i\geq m^{\alpha}M^{\gamma}\sum_{i=1}^{n}b_i^{\beta +1}+\beta \sum_{i=1}^{n}a_i^{\beta+1}$

(van khea):Let $a, x_i, b, \alpha, \beta, \gamma$ be positive real numbers such that:
$x_i\in (a, b), \forall{x_i\leq 1}; (i=1, 2, ..., n)$ and $\alpha-\beta+\gamma=1, \forall{\beta=min(\alpha, \beta, \gamma)}$.
$\displaystyle \frac{\alpha a+\gamma b}{a^{\alpha}b^{\gamma}}\geq \frac{1}{n}\biggl((\frac{x_1}{x_2})^{\beta}+(\frac{x_2}{x_3})^{\beta}+...+(\frac{x_n}{x_1})^{\beta}\biggl)$