Rad HAZU, Matematičke znanosti, Vol. 30 (2026), 93-109. \( \)

NOVEL VARIANTS OF HERMITE-HADAMARD INEQUALITIES FOR \((\alpha,\eta)-\)CONVEX FUNCTIONS OF \(1^{st}\) AND \(2^{nd}\) KINDS

Muhammad Bilal and Asif R. Khan

Department of Mathematics, University of Karachi, University Road, Karachi-75270, Pakistan
Department of Mathematics, Government Dehli Inter Science College, Hussianabad, Karachi, Pakistan
e-mail:mbilalfawad@gmail.com

Department of Mathematics, University of Karachi, University Road, Karachi-75270, Pakistan
e-mail:asifrk@uok.edu.pk

Abstract.   In this article, we aim to present generalized results related to the well-known Hermite-Hadamard dual inequality for two distinct types of \((\alpha, \eta)\) convex functions, employing various techniques such as Hölder's and Power mean inequalities. Consequently, both established and new results will be encompassed as special cases. Additionally, we intend to explore some relationships between our findings with well-known special means and trapezoidal formula.

2020 Mathematics Subject Classification.   26A46, 26A51, 26D07, 26D99

Key words and phrases.   Hermite-Hadamard inequalities, \((\alpha,\eta)-\)convex functions of the \(1^{st}\) kind, \((\alpha,\eta)-\)convex functions of the \(2^{nd}\) kind, Hölder's integral inequality, power mean integral inequality

Full text (PDF) (free access)

https://doi.org/10.21857/ypn4oc5739

References:

  1. M. A. Ali, Z. Zhang and M. Fečan, Generalization of Hermite-Hadamard-Mercer and trapezoid formula type inequalities involving the beta function, Rocky Mountain J. Math. 54 (2024), 331–341.
    MathSciNet    CrossRef

  2. R. P. Agarwal, A. R. Khan and S. Saadi, Integral results related to similarly separable vectors in separable Hilbert spaces, Foundations 2 (2022), 813–826.
    CrossRef

  3. M. I. Bhatti, M. Iqbal and M. Hussain, Hadamard-type inequalities for \(s\)-convex functions I, Punjab. Univ. J. Math. (Lahore) 41 (2009), 51–60.
    MathSciNet

  4. M. Bilal and A. R. Khan, Generalized Hermite-Hadamard type inequalities for twice differentiable \((r,s)\)-convex functions, Appl. Math. E-Notes 24 (2024), 350–361.
    MathSciNet

  5. M. Bilal, S. S. Dragomir and A. R. Khan, Generalized Hermite-Hadamard type inequalities for \((\alpha,\eta,\gamma,\delta)-p\) convex functions, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29 (2025), 145–186.
    MathSciNet    CrossRef

  6. M. Bilal and A. R. Khan, New generalized Hermite–Hadamard inequalities for \(p\)-convex functions in the mixed kind, Eur. J. Pure Appl. Math. 14 (2021), 863–880.
    MathSciNet    CrossRef

  7. M. Bilal, N. Irshad and A. R. Khan, New version of generalized Ostrowski-Grüss type inequality, Stud. Univ. Babeş-Bolyai Math. 66 (2021), 441–455.
    MathSciNet    CrossRef

  8. M. Bohner, H. Budak and H. Kara, Post-Quantum Hermite-Jensen-Mercer inequalities, Rocky Mountain J. Math. 53 (2023), 17–26.
    MathSciNet    CrossRef

  9. S. S. Dragomir, Hermite-Hadamard type inequalities for \(MN\)-convex functions, Aust. J. Math. Anal. Appl. 18 (1), (2021), Art. 1, 127pp.
    MathSciNet

  10. Ch. Hermite, Sur deux limites d'une intégrale définie, Mathesis 3 (1883), 82.

  11. A. Hassan and A. R. Khan, Generalized fractional Ostrowski type inequalities via \((\alpha,\beta,\gamma,\delta)\)-convex functions, Fract. Differ. Calc. 12 (2022), 13–36.
    MathSciNet    CrossRef

  12. Z. X. Mao, W. B. Zhang and J. F. Tian, On Hardy and Hermite-Hadamard inequalities for \(N\)-tuple diamond-alpha integral, Hacet. J. Math. Stat. 53 (2024), 667–689.
    MathSciNet

  13. D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers, Dordrecht, 1993.
    MathSciNet    CrossRef

  14. M. A. Noor, K. I. Noor, M. M. U. Awan and S. Khan, Fractional Hermite-Hadamard inequalities for some new classes of Godunova-Levin functions, Appl. Math. Inf. Sci. 8 (2014), 2865–2872.
    MathSciNet    CrossRef

  15. M. E. Özdemír, M. Avci and H. Kavurmaci, Hermite-Hadamard-type inequalities via (\(\alpha\), \(m\))-convexity, Comput. Math. App. 61 (2011), 2614–2620.
    MathSciNet    CrossRef

  16. S. Özcan, and I. Işcan, Some new Hermite-Hadamard type inequalities for \(s\)-convex functions and their applications, J. Inequal. App. (2019), Article 201.
    MathSciNet    CrossRef

  17. J. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.
    MathSciNet

  18. J. Pečarić, J. Barić, Lj. Kvesić and M. Ribičić Penava, Estimates on some quadrature rules via weighted Hermite-Hadamard inequality, Appl. Anal. Discrete Math. 16 (2022), 232–245.
    MathSciNet    CrossRef

  19. B. Y. Xi and F. Qi, Inequalities of Hermite-Hadamard type for extended \(s\)-convex functions and applications to means, J. Nonlinear Convex Anal. 16 (2015), 873–890.
    MathSciNet    CrossRef
Rad HAZU Home Page