Quasi-copulas with a given sub-diagonal section

As is well known, the Frechet–Hoeffding bounds are the best possible for both copulas and quasi-copulas: for every (quasi-)copula Q, max{x + y -1, 0}<= Q(x, y)<= min{x, y} for all x, y from [0, 1]. Sharper bounds hold when the (quasi-)copulas take prescribed values, e.g., along their diagonal or horizontal resp. vertical sections. Here we pursue two goals: first, we investigate construction methods for (quasi-)copulas with a given sub-diagonal section, i.e., with prescribed values along the straight line segment joining the points (x0, 0) and (1, 1 - x0) for x0 from ]0, 1[. Then, we determine the best-possible bounds for sets of quasi-copulas with a given sub-diagonal section.