Post Quantum Integral Inequalities of Hermite-Hadamard-Type Associated with Co-Ordinated Higher-Order Generalized Strongly Pre-Invex and Quasi-Pre-Invex Mappings

• Main Authors: Humaira Kalsoom, Saima Rashid, Muhammad Idrees, Farhat Safdar, Saima Akram, Dumitru Baleanu, Yu-Ming Chu
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Symmetry
• Subjects: quantum calculus; post quantum calculus hermite-hadamard type inequality; strongly pre-invex mappings; co-ordinated generalized higher-order strongly pre-invex mappings; co-ordinated generalized higher-order strongly quasi-pre-invex mappings
• Online Access: https://www.mdpi.com/2073-8994/12/3/443

By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard&#8217;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.