Differentiability of Functions on Spheres and Criterions of Convexity

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