In this paper, we study the concept of exponential convex functions with respect to s and prove Hermite-Hadamard type inequalities for the newly introduced this class of functions. In addition, we get some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential convex with respect to s. Our results coincide with the results obtained previously in special cases.