In this manuscript, we introduce and study the concept of $\\left( m_{1},m_{2}\\right) $-geometric arithmetically (GA) convex functions and their some algebric properties. In addition, we obtain Hermite-Hadamard type inequalities for the newly introduced this type of functions whose derivatives in absolute value are the class of $\\left( m_{1},m_{2}\\right) $ -GA-convex functions by using both well-known power mean and Hölder's integral inequalities.