  As a generalization of geodesic function, this paper introduces the notion of
geodesic $ \varphi_{E} $-convex function. Some properties of $ \varphi_{E}
$-convex function and geodesic $ \varphi_{E} $-convex function are established.
The concepts of geodesic $ \varphi_{E} $-convex set and $ \varphi_{E}
$-epigraph are equally given. The characterization of geodesic $ \varphi_{E}
$-convex functions in terms of their $ \varphi_{E} $-epigraphs are also
obtained.
