Mathematica Aeterna, Vol. 5, 2015, no. 3, 521 - 535 On Strongly m-Convex Functions Teodoro Lara Departamento de F´ ısica y Matem´ aticas. Universidad de Los Andes. Venezuela. tlara@ula.ve Nelson Merentes Escuela de Matem´ aticas. Universidad Central de Venezuela. Caracas. Venezuela nmerucv@gmail.com Roy Quintero Departamento de F´ ısica y Matem´ aticas. Universidad de Los Andes. Venezuela. rquinter@ula.ve Edgar Rosales Departamento de F´ ısica y Matem´ aticas. Universidad de Los Andes. Venezuela. edgarr@ula.ve Abstract The main purpose of this research is to introduce the definition of a strongly m-convex function. To achieve this goal, we generalize the well known notion of a strongly convex function by following a similar procedure to the one employed to generate the notion of functional m- convexity from the classical functional convexity. Several properties of this new class of functions are established as well as some inequalities of Jensen type in the discrete case. In the course of this study, we prove some additional and interesting results for the bigger class of m-convex functions. Mathematics Subject Classification: 26A51, 52A30 Keywords: m-convex function, strongly convex function, strongly m- convex function, Jensen type inequalities. 1 This research has been partially supported by Central Bank of Venezuela.