A new extension of quantum Simpson's and quantum Newton's type inequalities for quantum differentiable convex functions

Abstract

In this paper, we prove two identities involving quantum derivatives, quantum integrals, and certain parameters. Using the newly proved identities, we prove new inequalities of Simpson's and Newton's type for quantum differentiable convex functions under certain assumptions. Moreover, we discuss the special cases of our main results and obtain some new and existing Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities, and trapezoidal type inequalities.

Reversing Jensen’s Inequality for Information-Theoretic Analyses

“…This observation has spurred significant research efforts aimed at developing various versions of the so-called reverse Jensen inequality (RJI). A myriad of articles, including, but not limited to, [200,201,202,203,204,205,206,207,208,209], have delved into this topic, showcasing its rich and evolving landscape. In the majority of these works, the derived inequalities find practical applications in diverse fields.…”

Some Families of Jensen-like Inequalities with Application to Information Theory

“…[21,22] for inequalities via Chebychev and Chernoff bounds, ref. [23] for quantum Simpson's and quantum Newton's inequalities, and ref. [24] for new quantum Hermite-Hadamard-like inequalities.…”

Quantum Hermite-Hadamard Type Inequalities for Interval-Valued Functions

“…It is worth noting that, as a non-negligible and key branch of q-calculus, the study of quantum integral inequalities is a fresh and fascinating field of study. Some classical integral inequalities, such as Hölder, Hermite-Hadamard, Cauchy-Schwarz, Grüss, and Chebyshev inequalities were further expanded under the concept of q-calculus, see [2,4,5,12,13,14,38]. Especially, some further extensions of the q-Hermite-Hadamard (q-H-H) inequality for convex functions have been extensively investigated in [1,6,7,24,27,37].…”