THE AIM OF THIS WORK IS TO ESTABLISH SOME NEW ESTIMATES TO THE HERMITE-HADAMARD TYPE INEQUALITIES FOR (S, M) -CONVEX FUNCTIONS AND WE ESTABLISH SOME PROPERTIES OF H -CONVEX FUNCTIONS.

IN THIS ARTICLE, FIRST WE STATE THE NOTIONS OF METRIC CONVEXITY OF SETS IN THE SENSE OF MANGER [1] AND IN THE SENSE OF SOLTAN [5]. THEN WE REVIEW SOME PROPERTIES OF CONVEX SETS AND CONVEX HALLS IN THE SENSE OF SOLTAN. FINALLY, WE STUDY THE RELATION BETWEEN METRIC CONVEXITY AND GEOMETRIC PROPERTIES OF F-SPACES.

1. EXTENDED GDEA MODELAN EXTENSION OF GENERALIZED DATA ENVELOPMENT ANALYSIS (GDEA) MODEL HAS BEEN INTRODUCED IN THIS PAPER. IN THIS GENERALIZATION DIFFERENT CASES OF PRODUCTION POSSIBILITY SET (PPS) OF THE EXTENDED GENERALIZED DATA ENVELOPMENT ANALYSIS (EGDEA) MODEL ARE INVESTIGATED. THE PROPOSED EGDEA MODEL EVALUATES THE DECISION’S MAKING UNITS (DMUS) EITHER WITH FULL CONVEXITY ASSUMPTION IMPOSED ON INPUTS AND OUTPUTS (CCR AND BCC MODELS) OR WITHOUT CONVEXITY ASSUMPTION FOR EACH INPUT AND OUTPUT (FDH MODEL). WE HAVE PROPOSED A MODEL IN EGDEA, WHICH CAN TREAT EACH INPUT AND OUTPUT INDIVIDUALLY WITH RESPECT TO THE CONVEXITY ASSUMPTION. THE CONCEPT OF SELECTIVE CONVEXITY IN DATA ENVELOPMENT ANALYSIS (DEA) HAS BEEN UTILIZED IN THE PROPOSED EGDEA MODEL.

THE THEORY OF ICR (INCREASING AND CO-RADIANT) FUNCTIONS DEFINED ON ORDERED TOPOLOGICAL VECTOR SPACES HAS WELL BEEN DEVELOPED. IN THIS PAPER, WE PRESENT THE THEORY OF ICR-K (INCREASING AND CO-RADIANT OF DEGREE K) FUNCTIONS DEFINED ON AN ORDERED TOPOLOGICAL VECTOR SPACEX. WE FIRST GIVE A CHARACTERIZATION FOR ICR-K FUNCTIONS AND EXAMINE ABSTRACT CONVEXITY OF THIS CLASS OF FUNCTIONS. FINALLY, WE CHARACTERIZE SUPPORT SET AND SUBDIFFERENTIAL OF ICR-K FUNCTIONS.