New Inequalities for GA–h Convex Functions via Generalized Fractional Integral Operators with Applications to Entropy and Mean Inequalities

• Published in: Fractal and fractional 2024-12, Vol.8 (12), p.728
• Main Authors: Fahad, Asfand, Ali, Zammad, Furuichi, Shigeru, Butt, Saad Ihsan, Ayesha, Ayesha, Wang, Yuanheng
• Format: Article
• Language: English
• Subjects: arithmetic mean; Convex analysis; Entropy; Fractals; Fractional calculus; GA-h-convex functions; generalized Hadamard fractional integral operators; h-convex functions; Hermite–Hadamard inequality; Inequalities; Inequality; Investigations; Mathematical functions; Mercer inequality; Operators (mathematics); Upper bounds
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract8120728

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy.