Newton and Simpson Formula Type Inequalities Via Generalized Convexity in Quantum Calculus

Abstract

This study establishes Newton- and Simpson-type integral inequalities within the framework of generalized convexity in quantum calculus. By extending classical convexity concepts to the quantum calculus setting, new bounds are derived for differentiable functions defined on quantum time scales. The work develops refined integral inequalities using generalized convex functions, providing sharper estimates than their classical counterparts. Newton and Simpson-type formulas are adapted to incorporate q-calculus operators, and the resulting inequalities are analyzed for accuracy and convergence behavior. Applications of these results are discussed in numerical integration and approximation theory. The findings demonstrate that generalized convexity in quantum calculus offers powerful tools for improving classical numerical inequality estimates.