Unified Bounds for Quantum-Plank Integrals and Generalized Hermite-Hadamard Inequalities

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DOI: https://doi.org/10.28924/2291-8639-24-2026-97
Published: Apr 7, 2026
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Muhammad Ajmal, Muhammad Rafaqat, Waseela Bibi, Abdelgalal O.I. Abaker, Ghulam Farid, Unified Bounds for Quantum-Plank Integrals and Generalized Hermite-Hadamard Inequalities, Int. J. Anal. Appl., 24 (2026), 97.

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Muhammad Ajmal, Muhammad Rafaqat, Waseela Bibi, Abdelgalal O.I. Abaker, Ghulam Farid

Abstract

This paper investigates the boundedness of quantum-Planck \((q-h)\)-integral operators through novel extensions of classical inequalities, with particular emphasis on a broad spectrum of convexities, including standard convexity, \((\alpha, m)\)-convexity, \((s,m)\)-convexity, and related generalizations. Employing the versatile \((\alpha, \hbar-m)\)-convexity framework, we establish new Hermite-Hadamard type inequalities that not only unify existing approaches but also extend them to the setting of quantum calculus. Our results provide explicit upper and lower bounds for \((q-h)\)-integrals, delivering sharper refinements for special subclasses such as \((\alpha, \hbar-m)\)-p-convex functions. These developments encompass and generalize earlier quantum integral inequalities, thereby strengthening the theoretical foundation for stability, convergence, and approximation in quantum physics, combinatorics, and fractional modeling.

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