  As a generalization of geodesic function, in the present paper, we introduce
the notion of geodesic $\varphi$-convex function and deduce some basic
properties of $\varphi$-convex function and geodesic $\varphi$-convex function.
We also introduce the concept of geodesic $\varphi$-convex set and
$\varphi$-epigraph and investigate a characterization of geodesic
$\varphi$-convex functions in terms of their $\varphi$-epigraphs.
