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.
Highlights
In the study of quantum calculus, it is the non-limited analysis of calculus and it is recognized as q-calculus
Humaira et al [53] showed that the q1 q2 -Hermite-Hadamard type inequalities were resolved by the use of quantum calculus for the co-ordinated convex functions
We develop the theory of ( p1 p2, q1 q2 )-differentiable function and ( p1 p2, q1 q2 )-integral for two variables mappings over finite rectangles by using pre-invex set and let U = [ξ 1, ξ 1 + η1 (ξ 2, ξ 1 )] ×
Summary
In the study of quantum calculus, it is the non-limited analysis of calculus and it is recognized as q-calculus. The following Hermite-Hadamard type inequalities for co-ordinated convex functions on a rectangle from the plane R2 have been introduced by Dragomir [52]. Humaira et al [53] showed that the q1 q2 -Hermite-Hadamard type inequalities were resolved by the use of quantum calculus for the co-ordinated convex functions. Several novel versions of Hermite-Hadamard inequality are established that could be used to identify a uniform reflex ( p1 p2 , q1 q2 )-integral These innovations are a mixture of an identity-based auxiliary result that corresponds with co-ordenated generalized higher-order strongly pre-invex and quasi-pre-invex mappings. There are mathematical approximations of the new Definitions 10 and 11 are introduced for ( p1 p2 , q1 q2 )-differentiable function and ( p1 p2 , q1 q2 )-integral for two variables mappings over finite rectangles by using pre-invex set. An intriguing feature of the present investigation is that it has a potential connection with quantum mechanics and the special theory of relativity, see [54]
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