  We investigate how the type of Convexity of the Core function affects the
Csisz\'{a}r $f$-divergence functional. A general treatment for the type of
convexity has been considered and the associated perspective functions have
been studied. In particular, it has been shown that when the core function is
\rm{MN}-convex, then the associated perspective function is jointly
\rm{MN}-convex if the two scalar mean \rm{M} and \rm{N} are the same. In the
case where $\mathrm{M}\neq\mathrm{N}$, we study the type of convexity of the
perspective function. As an application, we prove that the \textit{Hellinger
distance} is jointly \rm{GG}-convex. As further applications, the matrix Jensen
inequality has been developed for the perspective functions under different
kinds of convexity.
