Jensen Type Inequality for L-Gap Convex Functions

JinYan Miao

doi:10.2478/amsil-2026-0006

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Jensen Type Inequality for L-Gap Convex Functions

Annales Mathematicae Silesianae

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By: JinYan Miao

Open Access

|Apr 2026

Abstract

The concept of l-gap convex functions is defined, which is more general than convex functions and allows some non-convex parts of the function. A Jensen type inequality is established and some examples are discussed. As an application and generalization, we prove that the majorization theorem also holds for l-gap convex functions. Then we use the conclusions from the above sections to establish a Hermite-Hadamard type inequality for l-gap convex functions.

References

M. Andrić and J.E. Pečarić, On (h, g; m)-convexity and the Hermite-Hadamard inequality, J. Convex Anal. 29 (2022), no. 1, 257–268.
Search in Google Scholar Back to article
A.J. Aw and N.A. Rosenberg, Bounding measures of genetic similarity and diversity using majorization, J. Math. Biol. 77 (2018), no. 3, 711–737.
Search in Google Scholar Back to article
N.S. Barnett, P. Cerone, and S.S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett. 22 (2009), no. 3, 416–421.
Search in Google Scholar Back to article
A. Belloni, T.Y. Liang, H. Narayanan, and A. Rakhlin, Escaping the local minima via simulated annealing: optimization of approximately convex functions, in: P. Grünwald et al. (eds.), Proceedings of the 28th Conference on Learning Theory, Proc. Mach. Learn. Res., 40, PMLR, 2015, pp. 240–265. Available at arXiv: 1501.07242.
Search in Google Scholar Back to article
J.M. Borwein and J.D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia Math. Appl., 109, Cambridge University Press, Cambridge, 2010.
Search in Google Scholar Back to article
M. Dehmer and F. Emmert-Streib (eds.), Quantitative Graph Theory: Mathematical Foundations and Applications, CRC Press, Boca Raton, 2015.
Search in Google Scholar Back to article
S.J. Dilworth, R. Howard, and J.W. Roberts, Extremal approximately convex functions and estimating the size of convex hulls, Adv. Math. 148 (1999), no. 1, 1–43.
Search in Google Scholar Back to article
S.J. Dilworth, R. Howard, and J.W. Roberts, A general theory of almost convex functions, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3413–3445.
Search in Google Scholar Back to article
S.S. Dragomir (ed.), Inequalities for Csiszár f-Divergence in Information Theory, RGMIA Monographs, Victoria University, 2000.
Search in Google Scholar Back to article
S.S. Dragomir, Hermite-Hadamard type inequalities for MN-convex functions, Aust. J. Math. Anal. Appl. 18 (2021), no. 1, Art. 1, 127 pp.
Search in Google Scholar Back to article
S.S. Dragomir and C.E.M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57 (1998), no. 3, 377–385.
Search in Google Scholar Back to article
S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
Search in Google Scholar Back to article
H.J. Greenberg and W.P. Pierskalla, A review of quasi-convex functions, Operations Res. 19 (1971), no. 7, 1553–1570.
Search in Google Scholar Back to article
G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, London, 1934.
Search in Google Scholar Back to article
D.H. Hyers and S.M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), no. 5, 821–828.
Search in Google Scholar Back to article
M.A. Khan, S.I. Bradanović, N. Latif, Ð. Pečarić, and J.E. Pečarić, Majorization Inequality and Information Theory. Selected Topics of Majorization and Applications, Monogr. Inequal., 16, ELEMENT, Zagreb, 2019.
Search in Google Scholar Back to article
A.R. Khan, J.E. Pečarić, M. Praljak, and S. Varošanec, General Linear Inequalities and Positivity. Higher Order Convexity, Monogr. Inequal., 12, ELEMENT, Zagreb, 2017.
Search in Google Scholar Back to article
J.C. Kuang, Applied Inequalities, 4th ed., (in Chinese), Shandong Science and Technology Press, Jinan, 2010.
Search in Google Scholar Back to article
T.Y. Li and R.Z. Zhang, Quantum speedups of optimizing approximately convex functions with applications to logarithmic regret stochastic convex bandits, in: S. Koyejo et al. (eds.), NIPS’22: Proceedings of the 36th International Conference on Neural Information Processing Systems, Curran Associates Inc., Red Hook, NY, 2022, Art. no. 228, pp. 3152–3164. Available at arXiv: 2209.12897.
Search in Google Scholar Back to article
A.W. Marshall, I. Olkin, and B.C. Arnord, Inequalities: Theory of Majorization and Its Applications, 2nd ed., Springer Ser. Statist., Springer, New York, 2011.
Search in Google Scholar Back to article
D.S. Mitrinović, J.E. Pečarić, and A.M. Fink, Classical and New Inequalities in Analysis, Math. Appl. (East European Ser.), 61, Springer Science+Business Media, Dordrecht, 1993.
Search in Google Scholar Back to article
K. Mosler, Majorization in economic disparity measures, Linear Algebra Appl. 199 (1994), 91–114.
Search in Google Scholar Back to article
C.T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 118 (1993), no. 1, 103–108.
Search in Google Scholar Back to article
C.P. Niculescu and L.-E. Persson, Convex Functions and Their Applications, CMS Books Math./Ouvrages Math. SMC, 23, Springer, New York, 2006.
Search in Google Scholar Back to article
J.C. Parnami and H.L. Vasudeva, On the stability of almost convex functions, Proc. Amer. Math. Soc. 97 (1986), no. 1, 67–70.
Search in Google Scholar Back to article
J.E. Pečarić, On some inequalities for functions with nondecreasing increments, J. Math. Anal. Appl. 98 (1984), no. 1, 188–197.
Search in Google Scholar Back to article
J.E. Pečarić, F. Proschan, and Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Engrg., 187, Academic Press, Inc., Boston, MA, 1992.
Search in Google Scholar Back to article
J.E. Pečarić and D. Zwick, n-convexity and majorization, Rocky Mountain J. Math. 19 (1989), no. 1, 303–311.
Search in Google Scholar Back to article
J. Ponstein, Seven kinds of convexity, SIAM Rev. 9 (1967), no. 1, 115–119.
Search in Google Scholar Back to article
T. Popoviciu, Les Fonctions Convexes, Actualités Sci. Indust., No. 992 [Current Scientific and Industrial Topics], Hermann & Cie, Paris, 1944.
Search in Google Scholar Back to article
A. Prékopa, Inequalities for discrete higher order convex functions, J. Math. Inequal. 4 (2009), no. 4, 485–498.
Search in Google Scholar Back to article
W.-L. Wang, Approaches to Prove Inequalities, (in Chinese), Harbin Institute of Technology Press, Harbin, 2011.
Search in Google Scholar Back to article
Ü. Yüceer, Discrete convexity: convexity for functions defined on discrete spaces, Discrete Appl. Math. 119 (2002), no. 3, 297–304.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2026-0006 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

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Language: English

Submitted on: Jan 12, 2025

Accepted on: Mar 10, 2026

Published on: Apr 4, 2026

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

26A51,

26D15

Related subjects:

Mathematics,

General mathematics

© 2026 JinYan Miao, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.

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