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Harremo(e)s obtained some new maximal inequalities for non-negative martingales.In this paper, we get some new maximal and minimal inequalities for non-negative demimartingales which generalize the results of Harremo(e)s. We also obtain an inequality for non-negative demimartingales which generalizes the result of Iksanov and Marynych. Finally we obtain a strong law of large numbers, strong growth rate and integrability of supremum for demimartingales which generalize and improve the result of Chow.