Abstract
In this paper, we establish some inequalities involving both \(h_1\)-convex functions and \(h_2\)-concave functions such that \( (h_1,h_2) \) is an increasing pair of functions. We apply the obtained results to Breckner’s s-convex functions, P-convex functions and Godunova–Levin functions. In order to unify the results, we introduce generalized s-convex functions of the second kind.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
Code availability
Not applicable.
References
Breckner, W.W., Trif, T.: Convex Functions and Related Functional Equations: Selected Topics. Cluj University Press, Cluj (2008)
Burtea, A.-M.: Two examples of weighted majorization. An. Univ. Craiova Ser. Mat. Inf. 37, 92–99 (2000)
Dragomir, S.S.: Inequalities of Hermite-Hadamard type for \(h\)-convex functions on linear spaces. Proyecc. J. Math. 34(4), 323–341 (2015)
Dragomir, S.S.: \(n\)-points inequalities of Hermite-Hadamard type for \(h\)-convex functions on linear spaces. Armen. J. Math. 8(1), 38–57 (2016)
Hardy, G.M., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Hudzik, H., Maligranda, L.: Some remarks on \(s\)-convex functions. Aequ. Math. 48, 100–111 (1994)
Karamata, J.: Sur une inégalité rélative aux fonctions convexes. Publ. Math. Univ. Belgrade 1, 145–148 (1932)
Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York (2011)
Micherda, B., Rajba, T.: On some Hermite-Hadamard-Fejér inequalities for \( (k, h) \)-convex functions. Math. Inequal. Appl. 15, 931–940 (2012)
Niezgoda, M.: Inequalities for convex functions and doubly stochastic matrices. Math. Inequal. Appl. 16, 221–232 (2013)
Niezgoda, M.: On \((k, h;m)\)-convex mappings and applications. Math. Inequal. Appl. 17, 665–678 (2014)
Niezgoda, M.: Inequalities for convex and concave functions and a new concept of majorization for two pairs of vectors. Results Math. 75(1), Article 34 (2020)
Pearce, C.E.M., Rubinov, A.M.: \(P\)-functions, quasi-convex functions and Hadamard-type inequalities. J. Math. Anal. Appl. 240, 92–104 (1999)
Sherman, S.: On a theorem of Hardy, Littlewood, Pólya, and Blackwell. Proc. Nat. Acad. Sci. USA 37, 826–831 (1957)
Varošanec, S.: On \(h\)-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)
Acknowledgements
The author wishes to thank an anonymous referee for his helpful suggestions improving the readability of the paper.
Funding
The author received no financial support for this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that he has no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Niezgoda, M. Inequalities of Sherman type involving both \(h_1\)-convex and \(h_2\)-concave functions with applications. Aequat. Math. 96, 1273–1283 (2022). https://doi.org/10.1007/s00010-022-00915-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-022-00915-0
Keywords
- h-convex function
- h-concave function
- s-convex function of the second kind
- P-function
- Godunova–Levin function
- Row stochastic matrix
- Column substochastic matrix