In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.

In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy.

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

In this study, we establish some new Hermite-Hadamard type inequalities for s-convex functions in the second sense using the post-quantum calculus. Moreover, we prove a new ( p , q ) -integral identity to prove some new Ostrowski type inequalities for ( p , q ) -differentiable functions. We also show that the newly discovered results are generalizations of comparable results in the literature. Finally, we give application to special means of real numbers using the newly proved inequalities.