Hermite-Hadamard inequalities for quantum integrals: A unified approach

• Published in: Applied mathematics and computation 2024-02, Vol.463, p.128345, Article 128345
• Main Authors: Cardoso, J.L., Shehata, Enas M.
• Format: Article
• Language: English
• Subjects: Convex function; General quantum difference operator; Hermite-Hadamard inequalities; β-derivative; β-Hermite-Hadamard inequalities; β-integral
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.128345

Let s0 be a fixed point of a strictly increasing continuous function β. Hamza et al. introduced the quantum operator Dβ[g](θ):=g(β(θ))−g(θ)β(θ)−θ, θ≠s0 and Dβ[g](s0):=g′(s0) if θ=s0. For specific choices of the function β one obtains the known Jackson q-operator Dq as well as the Hahn quantum operator Dq,ω. Regarding its inverse operator, the β-integral, we establish the corresponding β-Hermite-Hadamard inequalities. Among others, we also obtain the Hermite-Hadamard type inequalities for the Jackson q-integral, the Nörlund integral and for the Jackson-Thomae (or Jackson-Nörlund) q,ω-integral. •The main result is Theorem 3.1: it establishes the Hermite-Hadamard inequalities for the β-integral.•Subsection 3.3 comprises, as corollaries, the Hermite-Hadamard inequalities for all known quantum integrals.•Section 4 contains an application to special means and also a statistic interpretation.