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We derive a new theoretical lower bound for the expected supremum of drifted fractional Brownian motion with Hurst index 
$H\in(0,1)$ over a (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 in the general case and give an explicit expression for the derivative of the expected supremum at 
$H=\tfrac{1}{2}$ in the sense of Bisewski, Dȩbicki and Rolski (2021).
