Abstract: In this paper, one important type of generalized convex functions termed as strongly quasi $\alpha$-preinvex functions is studied. The relationships among strongly quasi $\alpha$-preinvex functions and quasi $\alpha$-preinvex functions, and strictly quasi $\alpha$-preinvex functions and semistrictly quasi $\alpha$-preinvex functions have been discussed. Three important theorems of properties have been obtained for it on the middle point strongly quasi $\alpha$-preinvexity. We also present two important applications in terms of strongly quasi $\alpha$-preinvex functions in mathematical programming. These results complete the researches for the strongly quasi $\alpha$-preinvex functions to some extent.

Key words: strongly quasi $\alpha$-preinvex functions, quasi $\alpha$-preinvex functions, strictly quasi $\alpha$-preinvex functions, semistrictly quasi $\alpha$-preinvex functions