  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^* (\tau)]^p\,d\tau \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 $$ (T_{u,b} g)(t) : =
\sup_{t \le \tau < \infty} \frac{u(\tau)}{B(\tau)} \int_0^{\tau}
g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ 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 $M_{\phi,\Lambda^{\alpha}(b)}$,
defined with $0 < \alpha < \infty$ and functions $b,\,\phi: (0,\infty)
\rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by
\begin{equation*} M_{\phi,\Lambda^{\alpha}(b)}f(x) : = \sup_{Q \ni x} \frac{\|f
\chi_Q\|_{\Lambda^{\alpha}(b)}}{\phi (|Q|)}, \qquad x \in {\mathbb R}^n,
\end{equation*} from ${\operatorname{G\Gamma}}(p_1,m_1,v)$ into
${\operatorname{G\Gamma}}(p_2,m_2,w)$. Here $\Lambda^{\alpha}(b)$ and
${\operatorname{G\Gamma}}(p,m,w)$ are the classical and generalized Lorentz
spaces, respectively.
