It is known that there is a constant c > 0 such that for every sequence x1, x2, . . . in [0, 1) we have for the star discrepancy $D_N^* $ of the first N elements of the sequence that $ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c∗ be the supremum of all such c with this property. We show c∗ > 0.065664679 . . . , thereby slightly improving the estimates known until now.