The Hermite-Hadamard type inequality has been extended to some classes of convex functions and composite convex functions. In this paper, we refine the definition of composite-ɸ−1 convex function and definition of m-convex function to define the composite-ɸ−1 m-convex functions. From the definition, we obtain some inequalities related to GA-m-convexity, HA-m-convexity, p-m-convexity and LogExp m-convexity. Next, k-composite-ɸ−1 m-convex function is defined. Some inequalities of Hermite-Hadamard type for composite-ɸ−1 m-convex functions and k-composite-ɸ−1 m-convex functions are obtained. From the inequalities obtained, we provide some applications for GG, AG, HA and HH convex functions.