Differentiability of Functions on Spheres and Criterions of Convexity

Some basic concepts for functions defined on subsets of the unit sphere, such as s-directional derivative, s-gradient and s-Gateaux s-Frechet differentiability etc, are introduced investigated. These different from usual ones Euclidean spaces, however, results obtained here very similar. Then, applications, we provide some criterions s-convexity spheres which improvements or refinements known results.