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.

In this paper we investigate conditions that lower semicontinuous functions on normed linear space to be E-CONVEX functions and then present some related results.

IN THIS PAPER, WE FIRST INVESTIGATE THE OPTIMIZATION OF CONVEX FUNCTIONS BY USING LAGRANGE MULTIPLIERS AND SADDLE POINTS.NEXT, WE PRESENT AN ALGORITHM FOR FINDING THE GLOBAL MAXIMIZERS OF THE CONSTRAINED NON-POSITIVE VALUED ICR (INCREASING AND CO-RADIANT) FUNCTIONS OVER THE UNIT SIMPLEX BY USING THE GLOBAL MAXI-MIZERS OF INCREASING AND POSITIVELY HOMOGENEOUS (IPH) FUNCTIONS.ALSO, WE GIVE SOME NUMERICAL EXPERIMENTS. FINALLY, NON-POSITIVE VALUED AFFINE ICR FUNCTIONS ARE DEFINED IN THE FRAMEWORK OF ABSTRACT CONVEXITY. THE BASIC PROPERTIES OF THIS CLASS OF FUNCTIONS SUCH AS SUPPORT SET AND SUB DIFFERENTIAL ARE PRESENTED. AS AN APPLICATION, WE GIVE OPTIMALITY CONDITIONS FOR THE GLOBAL MINIMUM OF THE DIFFERENCE OF TWO STRICTLY NON-POSITIVE VALUED AFFINE ICR FUNCTIONS.

SOME BASIC INEQUALITIES RELATED TO H-CONVEX FUNCTIONS ARE PROVED. ALSO WE INVESTIGATE SOME CHARACTERIZATION THEOREMS ABOUT H-CONVEX FUNCTIONS WITH NEW SETTINGS. FURTHERMORE WE OBTAINSOME INEQUALITIES RELATED TO THIS CLASS OF FUNCTIONS.