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New quantum boundaries for quantum Simpson's and quantum Newton's type inequalities for preinvex functions
(Springer, 2021) Ali, Muhammad Aamir; Abbas, Mujahid; Budak, Hüseyin; Agarwal, Praveen; Murtaza, Ghulam; Chu, Yu-Ming
In this research, we derive two generalized integral identities involving the q kappa 2-quantum integrals and quantum numbers, the results are then used to establish some new quantum boundaries for quantum Simpson's and quantum Newton's inequalities for q-differentiable preinvex functions. Moreover, we obtain some new and known Simpson's and Newton's type inequalities by considering the limit q -> 1- in the key results of this paper.
Post-quantum Hermite-Hadamard type inequalities for interval-valued convex functions
(Springer, 2021) Ali, Muhammad Aamir; Budak, Hüseyin; Murtaza, Ghulam; Chu, Yu-Ming
In this research, we introduce the notions of (p, q)-derivative and integral for interval-valued functions and discuss their fundamental properties. After that, we prove some new inequalities of Hermite-Hadamard type for interval-valued convex functions employing the newly defined integral and derivative. Moreover, we find the estimates for the newly proved inequalities of Hermite-Hadamard type. It is also shown that the results proved in this study are the generalization of some already proved research in the field of Hermite-Hadamard inequalities.