  We substantially refine asymptotic logarithmic upper bounds produced by
Svante Janson (2015) on the right tail of the limiting QuickSort distribution
function $F$ and by Fill and Hung (2018) on the right tails of the
corresponding density $f$ and of the absolute derivatives of $f$ of each order.
For example, we establish an upper bound on $\log[1 - F(x)]$ that matches
conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order
$(\log x)^2$; the corresponding order for the Janson (2015) bound is the lead
order, $x \log x$.
  Using the refined asymptotic bounds on $F$, we derive right-tail large
deviation (LD) results for the distribution of the number of comparisons
required by QuickSort that substantially sharpen the two-sided LD results of
McDiarmid and Hayward (1996).
