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

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 opera...

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Bibliographic Details
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
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Summary: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.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract8120728