Hermite-Hadamard's like inequalities via symmetric quantum calculus

It is evident that several inequalities have been explored via diverse strategies to acquire more reliable, robust, and accurate bounds of mathematical quantities. Although the role of convexity and generalized calculus is unprecedented in the development of both dynamical and analytical inequalities. Classical derivative and integral operators failed to investigate various classes of mappings. Symmetric calculus provides the framework to study the several classes of non-differentiable mappings. In this manuscript, we investigate the Hermite-Hadamard's and error inequalities of quadrature schemes leveraging the concepts of convex mapping and left and right symmetric operators. We start by proving various Hermite-Hadamard's like inequalities involving double, triple symmetric quantum integrals and a unified identity based on symmetric quantum differentiable mappings. Afterwards, we construct numerous integral inequalities corresponding with different quadrature schemes through convex mapping and elementary concepts from the theory of inequalities. To ensure the effectiveness and applicability of proposed results, we deduce several special scenarios, comparative analyses with previously explored results, and applications to linear combinations of means and composite quadrature schemes.