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For the fractional Brownian motion $B^H$ with the Hurst parameter value $H$in (0,1/2), we derive new upper and lower bounds for the difference between theexpectations of the maximum of $B^H$ over [0,1] and the maximum of $B^H$ overthe discrete set of values $ in^{-1},$ $i=1,\ldots, n.$ We use these results toimprove our earlier upper bounds for the expectation of the maximum of $B^H$over $[0,1]$ and derive new upper bounds for Pickands' constant.

New and refined bounds for expected maxima of fractional Brownian motion