Inequalities for weighted Hardy operators in weighted variable exponent Lebesgue space with $0 < p(x) < 1$

Weighted inequalities are proved for the weighted Hardy operators and the weighted dual of the classical Hardy operator acting from one weighted variable exponent Lebesgue space $L_{p(.),\omega_1} (0,\infty)$ to another weighted variable exponent Lebesgue space $L_{p(.),\omega_2} (0,\infty)$ for $0 < p(x) \leqslant q(x) < 1$.