Some integral inequalities of Hermite--Hadamard type for $(\beta,F)$-log-convex functions

Year 2025, Volume: 8 Issue: 3, 156 - 164, 15.09.2025

Yan Wang , Ximin Liu , Bo-yan Xi , Feng Qi

https://doi.org/10.33205/cma.1733628

Abstract

In the paper, the authors introduce the notion of $(\beta,F)$-log-convex functions, give an example of the $(\beta,F)$-log-convex functions, and, by virtue of two known integral identities, establish several integral inequalities of the Hermite--Hadamard type for $(\beta,F)$-log-convex functions.

Keywords

integral inequality of the Hermite--Hadamard type , $(\beta , F)$-log-convex function , example , integral identity

References

• M. Adamek: On a problem connected with strongly convex functions, Math. Inequal. Appl., 19(4) (2016), 1287–1293.
• M. Adamek: On Hermite–Hadamard type inequalities for F-convex functions, J. Math. Inequal., 14(3) (2020), 867–874.
• M. Adamek: On F-convex functions, J. Convex Anal., 28(3) (2021), 761–770.
• S. S. Dragomir, R. P. Agarwal: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
• S. S. Dragomir, C. E. M. Pearce: Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2002).
• U. S. Kirmaci: Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147(1) (2004), 137–146.
• N. Merentes, K. Nikodem: Remarks on strongly convex functions, Aequationes Math., 80(1-2) (2010), 193–199.
• C. E. M. Pearce, J. Peˇcari´c: Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55.
• J. E. Peˇcari´c, F. Proschan, and Y. L. Tong: Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston (1992).
• B. T. Polyak: Existence theorems and convergence of minimizing sequences in extremum problems with restictions, Soviet Math. Dokl., 7 (1966), 72–75.
• Y. Shuang, F. Qi: Integral inequalities of Hermite–Hadamard type for GA-F-convex functions, AIMS Math., 6(9) (2021), 9582–9589.
• Y. Wang, X.-M. Liu, and B.-N. Guo: Several integral inequalities of the Hermite–Hadamard type for s-(β, F)-convex functions, ScienceAsia, 49 (2) (2023), 200–204.