Several integral inequalities for strongly $(\alpha, m)$-convex functions with modulus

Year 2026, Volume: 9 Issue: 1, 1 - 8, 06.03.2026

Bo-Yan Xi , Tian-Yu Zhang , Feng Qi

https://doi.org/10.33205/cma.1868426

https://izlik.org/JA67SX84KZ

Abstract

In the paper, the authors introduce the concept of strongly $(\alpha, m)$-convex function with modulus, establish several new integral identities, and, by virtue of these newly-identities and H\"older's inequality, present some new integral inequalities of the Hermite--Hadamard type for strongly $(\alpha, m)$-convex functions with modulus.

Keywords

Hermite--Hadamard's inequality , strongly (α , m)-convex function , Hölder inequality

References

• M. K. Bakula, M. E. Özdemir and J. Peˇcari´c: Hadamard type inequalities for m-convex and (α,m)-convex functions, J. Inequal. Pure Appl. Math., 9 (4) (2008), Article ID: 96.
• S. S. Dragomir: On some new inequalities of Hermite–Hadamard type for m-convex functions, Tamkang J. Math., 33 (1) (2002), 45–55.
• 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 Type Inequalities and Applications, RGMIA Monographs, Victoria University (2000).
• S. S. Dragomir, G. Toader: Some inequalities for m-convex functions, Studia Univ. Babe¸s-Bolyai Math., 38 (1) (1993), 21–28.
• D. Y. Hwang: Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables, Appl. Math. Comput., 217 (23) (2011), 9598–9605.
• V. G. Mihe¸san: A generalization of the convexity, Seminar on Functional Equations, Approx. Convex, Cluj-Napoca (1993).
• C. P. Niculescu, L.-E. Persson: Convex Functions and Their Applications, CMS Books in Mathematics, 23, Springer-Verlag, New York (2005).
• B. T. Polyak: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 72–75.
• G. Toader: Some generalizations of the convexity, Proc. Col. Approx. Opt., Univ. Cluj-Napoca, Cluj-Napoca, (1985), 329–338.
• Y. Wang, X.-M. Liu, B.-Y. Xi and F. Qi: Integral inequalities of Hermite–Hadamard type for (α,m)-GA-F-convex functions, Filomat 39 (28) (2025), 10161–10170.
• B.-Y. Xi, R.-F. Bai and F. Qi: Hermite–Hadamard type inequalities for m-and (α,m)-geometrically convex functions, Aequationes Math., 84 (3) (2012), 261–269.
• B.-Y. Xi, D.-D. Gao and F. Qi: Integral inequalities of Hermite–Hadamard type for (α, s)-convex and (α, s,m)-convex functions, Italian J. Pure Appl. Math., 44 (2020), 499–510.