In this paper we characterize the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^x \big[ T_{u,b}f^* (t)\big]^r\,dt\bigg)^{\frac{q}{r}} w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (��)]^p\,d��\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} for $1 < m < p \le r < q < \infty$ or $1 < m \le r < \min\{p,q\} < \infty$, where $w$ and $v$ are weight functions on $(0,\infty)$. The inequality is required to hold with some positive constant $C$ for all measurable functions defined on measure space $({\mathbb R}^n,dx)$. Here $f^*$ is the non-increasing rearrangement of a measurable function $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, whish is defined for a measurable non-negative function $f$ on $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{t \le ��< \infty} \frac{u(��)}{B(��)} \int_0^�� g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are two weight functions on $(0,\infty)$ such that $u$ is continuous on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$. At the end of the paper, as an application of obtained results, we calculate the norm of the generalized maximal operator $M_{��,��^��(b)}$, defined with $0 < ��< \infty$ and functions $b,\,��: (0,\infty) \rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by \begin{equation*} M_{��,��^��(b)}f(x) : = \sup_{Q \ni x} \frac{\|f ��_Q\|_{��^��(b)}}{��(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from ${\operatorname{G��}}(p_1,m_1,v)$ into ${\operatorname{G��}}(p_2,m_2,w)$. Here $��^��(b)$ and ${\operatorname{G��}}(p,m,w)$ are the classical and generalized Lorentz spaces, respectively.

41 pages