In this paper, we introduce and study the concept of n-fractional polynomial P-functions and establish Hermite-Hadamard’s inequalities for this type of functions. In addition, we obtain some new Hermite-Hadamard type inequalities for functions whose first derivative in absolute value is n-fractional polynomial P-functions by using Hölder and power-mean integral inequalities. We also extend our initial results to functions of several variables. Next, we point out some applications of our results to give estimates for the approximation error of the integral the function in the trapezoidal formula and for some inequalities related to special means of real numbers.