  We derive a new theoretical lower bound for the expected supremum of drifted
fractional Brownian motion with Hurst index $H\in(0,1)$ over (in)finite time
horizon. Extensive simulation experiments indicate that our lower bound
outperforms the Monte Carlo estimates based on very dense grids for
$H\in(0,\tfrac{1}{2})$. Additionally, we derive the Paley-Wiener-Zygmund
representation of a Linear Fractional Brownian motion and give an explicit
expression for the derivative of the expected supremum at $H=\tfrac{1}{2}$ in
the sense of recent work by Bisewski, D\k{e}bicki & Rolski (2021).
