The concept of $f$-divergences was introduced by Csiszár in 1963 as measures of the ‘hardness’ of a testing problem depending on a convex real valued function $f$ on the interval $[0,\infty )$. The choice of this parameter $f$ can be adjusted so as to match the needs for specific applications. The definition and some of the most basic properties of $f$-divergences are given and the class of ${\chi }^{\alpha }$-divergences is presented. Ostrowski’s inequality and a Trapezoid inequality are utilized in order to prove bounds for an extension...

Nelsen et al. [20] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. Tankov [25] generalizes this work by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0; 1]2 are known. He shows that they are quasi-copulas and not necessarily copulas. Tankov [25] and Bernard et al. [3] both give sufficient conditions for these bounds to be copulas. In this note we give weaker...