Hardy type inequalities in classical and grand Lebesgue spaces $L_{p)}$, $0<p\leqslant 1$, for quasi-monotone functions

In 2020 Rovshan A. Bandaliyev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces $L_{p)} (0,1)$, $0<p\leqslant 1$. In particular,they established similar results for the Hardy operator in weighted classical Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space $L_{p) } (0,1)$ is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. we establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces $L_{p)}$, $w(0,1)$ $0<p\leqslant 1$. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces $L_{p,w} (0,1)$, $0<p<1$ for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliyev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.