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We discuss $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities ofweighted maximal operators of $T$ means with respect to the Vilenkin systemswith monotone coefficients, considered in \cite{tut4} and prove that theseresults are the best possible in a special sense. As applications, both somewell-known and new results are pointed out.

Sharp $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities of weighted maximal operators of $T$ means with respect to Vilenkin systems