On generalized uniformly tapered matrices (Q1209411)

A vector \(y\) is said to \(c\)-majorize \(x\) if \(c=Sc\) and \(x=Sy\) for some stochastic matrix \(S\). The well-known characterization of \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya} [Some simple inequalities satisfied by convex functions. Mess. Math. 58, 145-152 (1929; JFM 55.0740.04)] generalizes to this setting, where \(x\), \(y\) are such that the ratios \(x_ i/c_ i\) and \(y_ i/c_ i\) are nonincreasing. Here it is shown that the matrix \(S\) can be chosen within the convex hull of a finite set of matrices associated with partitions of the coordinates, and this convex hull is described.