On $q$-Hermite–Hadamard–Mercer and midpoint-Mercer inequalities for general convex functions with their computational analysis

This paper establishes some new inequalities of Hermite–Hadamard–Mercer type for $s$-convex functions in the framework of $q$-calculus and classical calculus. Some new $q$-midpoint-Mercer type inequalities for the $q$-differentiable $s$-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and $s$-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for $s$-convex functions in $q$-calculus.