(q −h)-Hermite-Hadamard, Midpoint, and Trapezoidal Inequalities  for Convex Functions

Iram  Javed; Arslan  Razaq; Prof. Juan Napoles

doi:10.5269/bspm.77047

Iram Javed Beihang University of Aeronautics and Astronautics https://orcid.org/0000-0002-5212-4422
Arslan Razaq China University of Geosciences https://orcid.org/0009-0001-9843-3300
Prof. Juan Napoles

Resumo

This paper establishes transformative advances in quantum calculus by introducing a comprehensive framework for (q − h)-generalized integral inequalities. We pioneer three foundational contributions to the field: First, we derive a novel bilateral (q − h)-Hermite-Hadamard inequality through an innovative synthesis of left and right quantum integrals, substantially generalizing classical integral inequalities. Second, we develop sharp midpoint-type inequalities with explicit error bounds for (q − h)-differentiable convex functions, employing strategic applications of Holder’s inequality and power mean inequalities to quantify approximation accuracy. Third, we construct advanced trapezoidal estimators incorporating
quantum-calculus modifications that systematically account for parameterized displacements h. Through rigorous constructive proofs and numerical validation, we demonstrate that our unified framework naturally contains classical calculus as the limiting case lim q→1,h→0 achieves tighter error bounds compared to existing q-calculus results through optimized (q,h)-coupling and enables precision-tunable approximations via flexible parameterization of q and h. These breakthroughs significantly extend the operational calculus for convex functions in non-Newtonian analysis, with immediate applications in quantum probability measures, fractional variational optimization, and deformed mathematical physics models requiring
non-uniform discretization schemes.

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Biografia do Autor

Iram Javed, Beihang University of Aeronautics and Astronautics

School of Mathematical Sciences

Arslan Razaq, China University of Geosciences

School of Mathematics and Physics

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