Given: Strictly positive numbers, $$ m_1, m_2, n_1, n_2, x, y, g, k $$ which satisfy the following properties: $$ m_1 + m_2 = n_1 + n_2 $$ $$ \frac{m_2}{m_1} \le k \le \frac{n_2}{n_1} \le 1 $$ $$ \frac{m_2}{n_2} \le 1 \le \frac{m_1}{n_1} $$ $$ 1 \le g $$ $$ \frac{xm_2 + yn_2}{xm_1 + yn_1} < k < 1. $$
Show: For all feasible values, we have $$ \frac{xm_2^g + yn_2^g}{xm_1^g+yn_1^g} < k. $$