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

Sharp $\left( H_{p},L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients