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 of the set of $f$-divergences. The class of $\chi ^{\alpha }$-divergences and four further classes of $f$-divergences are used in order to investigate limitations and strengths of the inequalities derived.