A class ofG-semipreinvex functions, which are some generalizations of the semipreinvex functions, and theG-convex functions, is introduced. Examples are given to show their relations amongG-semipreinvex functions, semipreinvex functions andG-convex functions. Some characterizations ofG-semipreinvex functions are also obtained, and some optimality results are given for a class ofG-semipreinvex functions. Ours results improve and generalize some known results.