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In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness. [ABSTRACT FROM AUTHOR]

In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities. [ABSTRACT FROM AUTHOR]

In this study, first we establish a p , q -integral identity involving the second p , q -derivative, and then, we use this result to prove some new midpoint-type inequalities for twice- p , q -differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature. [ABSTRACT FROM AUTHOR]