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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.
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Kalsoom et al. (2020) studied this question.
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