Refined solution to upper-bound problem for the expectations of order statistics from decreasing density on the average distributions

Rychlik [Metrika 77, 539–557, 2014] described sharp upper negative bounds for the expectations of low-rank order statistics, centered about the population mean and measured in the mean absolute deviation from the mean units, for the i.i.d. sequences with common distribution possessing decreasing density function on the average. The bounds coincide with the negatives of maximal values of complicated functions on the unit interval. Here, we provide more precise solutions to the maximization problems.