  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 the
expectations of the maximum of $B^H$ over [0,1] and the maximum of $B^H$ over
the discrete set of values $ in^{-1},$ $i=1,\ldots, n.$ We use these results to
improve our earlier upper bounds for the expectation of the maximum of $B^H$
over $[0,1]$ and derive new upper bounds for Pickands' constant.
