IN THIS PAPER, WE INVESTIGATE THE RELATION BETWEEN NONINCREASING CONCAVE, HARMONICALLY CONCAVE AND (A, M) -convex FUNCTIONS. THE AUTHOR INTRODUCED CONDITIONS THAT UNDER THEM, IT HAS SHOWN THE CLOSE TO SET OF M-convex FUNCTIONS, (A, M) -convex AND HARMONICALLY (A, M) -convex FUNCTIONS WITH RESPECT TO SOME ALGEBRAIC ACTIONS AS ADDITION AND SUBTRACTION ON THEIR DOMAIN. ALSO, WE ESTABLISH THAT SOME OF MENTIONED FUNCTIONS SET IS CLOSED UNDER THE COMBINATION ACTION. IN FINALLY, WE OBTAIN SOME RESULTS FOR (A, M) -convex, (B, M) -convex AND (AB, M)-convexN FUNCTIONS.