Abstract:
By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard's type inequality and conclude explicit bounds for two new definitions of (p(1)p(2), q(1)q(2))-differentiable function and (p(1)p(2), q(1)q(2))-integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for (p(1)p(2), q(1)q(2))-integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for (p(1)p(2), q(1)q(2))-differentiable utilizing new definitions of generalized higher-order strongly pre-invex and quasi-pre-invex mappings. It is to be acknowledged that this research study would develop new possibilities in pre-invex theory, quantum mechanics and special relativity frameworks of varying nature for thorough investigation.
