| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-differentiable function and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-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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