  Smooth Csisz\'ar $f$-divergences can be expressed as integrals over so-called
hockey stick divergences. This motivates a natural quantum generalization in
terms of quantum Hockey stick divergences, which we explore here. Using this
recipe, the Kullback-Leibler divergence generalises to the Umegaki relative
entropy, in the integral form recently found by Frenkel. We find that the
R\'enyi divergences defined via our new quantum $f$-divergences are not
additive in general, but that their regularisations surprisingly yield the Petz
R\'enyi divergence for $\alpha < 1$ and the sandwiched R\'enyi divergence for
$\alpha > 1$, unifying these two important families of quantum R\'enyi
divergences. Moreover, we find that the contraction coefficients for the new
quantum $f$ divergences collapse for all $f$ that are operator convex,
mimicking the classical behaviour and resolving some long-standing conjectures
by Lesniewski and Ruskai. We derive various inequalities, including new reverse
Pinsker inequalites with applications in differential privacy and also explore
various other applications of the new divergences.
