  In this paper we prove and discuss some new $\left( H_{p},L_{p}\right)$ type
inequalities of maximal operators of $T$ means with respect to the Vilenkin
systems with monotone coefficients. We also apply these inequalities to prove
strong convergence theorems of such $T$ means. We also show that these results
are the best possible in a special sense. As applications, both some well-known
and new results are pointed out.
