Favard-Type Inequality on Symmetric Quantum Calculus

In this paper, we establish a fundamental lemma based on symmetric quantum monotonicity within the framework of symmetric quantum calculus. Using this lemma, we derive the symmetric quantum Favard inequality and its Berwald-type extensions, together with generalizations to products of multiple functions and broader concavity classes. Numerical examples and comparative analyses demonstrate that the upper bounds obtained in the symmetric quantum framework are closer to the true integral values than those from the asymmetric quantum setting. This work establishes, for the first time, a complete theory of Favard–Berwald-type inequalities in symmetric quantum calculus, filling a gap in the literature and providing new analytical tools for quantum optimization and discrete integral estimation.