Hermite–Hadamard type inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions

Chun-Ying He; Aying Wan; Bai-Ni Guo; Chun-Ying He; Aying Wan; Bai-Ni Guo

doi:10.3934/math.2023869

AIMS Mathematics

2023, Volume 8, Issue 7: 17027-17037. doi: 10.3934/math.2023869

Previous Article Next Article

Research article Special Issues

Hermite–Hadamard type inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions

Chun-Ying He ¹,
Aying Wan ^{1
,
,},
Bai-Ni Guo ^{2,3
,
,}

1.
School of Mathematics and Statistics, Hulunbuir University, Hulunbuir 021008, Inner Mongolia, China
2.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
3.
Independent researcher, Dallas, TX 75252-8024, USA

Received: 08 March 2023 Revised: 27 April 2023 Accepted: 09 May 2023 Published: 16 May 2023
MSC : 26A51, 26D15, 26D20, 26E60, 41A55

In this paper, the authors define the notion of harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, establish a new integral identity, present some new Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, and derive some known results.
Citation: Chun-Ying He, Aying Wan, Bai-Ni Guo. Hermite–Hadamard type inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions[J]. AIMS Mathematics, 2023, 8(7): 17027-17037. doi: 10.3934/math.2023869

Related Papers:

Abstract

In this paper, the authors define the notion of harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, establish a new integral identity, present some new Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, and derive some known results.

References

[1]	O. Y. Mohammed Alabdali, A. Guessab, On the approximation of strongly convex functions by an upper or lower operator, Appl. Math. Comput., 247 (2014), 1129–1138. http://dx.doi.org/10.1016/j.amc.2014.09.007 doi: 10.1016/j.amc.2014.09.007
[2]	W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen (German), Publications de l'Institut Mathématique, 23 (1978), 13–20.
[3]	S. S. Dragomir, On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775–788. http://dx.doi.org/10.11650/twjm/1500574995 doi: 10.11650/twjm/1500574995
[4]	S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite–Hadamard type inequalities and applications, RGMIA Monographs, Victoria University, 2000. Available from: http://rgmia.org/monographs/hermite_hadamard.html
[5]	C.-Y. He, B.-Y. Xi, B.-N. Guo, Inequalities of Hermite–Hadamard type for extended harmonically $(s, m)$-convex functions, Miskolc Math. Notes, 22 (2020), 245–258. http://dx.doi.org/10.18514/MMN.2021.3080 doi: 10.18514/MMN.2021.3080
[6]	H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aeq. Math., 48 (1994), 100–111. http://dx.doi.org/10.1007/BF01837981 doi: 10.1007/BF01837981
[7]	V. G. Miheşan, A generalization of the convexity (Romania), Proceedings of Seminar on Functional Equations, 1993.
[8]	F. Qi, Z.-L. Wei, Q. Yang, Generalizations and refinements of Hermite–Hadamard's inequality, Rocky Mountain J. Math., 35 (2005), 235–251. http://dx.doi.org/10.1216/rmjm/1181069779 doi: 10.1216/rmjm/1181069779
[9]	G. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization (Cluj-Napoca), 1985,329–338.
[10]	B.-Y. Xi, R.-F. Bai, F. Qi, Hermite–Hadamard type inequalities for the $m$- and $(\alpha, m)$-geometrically convex functions, Aequat. Math., 84 (2012), 261–269. http://dx.doi.org/10.1007/s00010-011-0114-x doi: 10.1007/s00010-011-0114-x
[11]	B.-Y. Xi, D.-D. Gao, F. Qi, Integral inequalities of Hermite–Hadamard type for $(\alpha, s)$-convex and $(\alpha, s, m)$-convex functions, Ital. J. Pure Appl. Math., 44 (2020), 499–510.
[12]	B.-Y. Xi, C.-Y. He, F. Qi, Some new inequalities of the Hermite–Hadamard type for extended $((s_1, m_1)$-$(s_2, m_2))$-convex functions on co-ordinates, Cogent Math., 3 (2016), 1267300. http://dx.doi.org/10.1080/23311835.2016.1267300 doi: 10.1080/23311835.2016.1267300
[13]	B.-Y. Xi, F. Qi, Hermite–Hadamard type inequalities for geometrically $r$-convex functions, Stud. Sci. Math. Hung., 51 (2014), 530–546. http://dx.doi.org/10.1556/sscmath.51.2014.4.1294 doi: 10.1556/sscmath.51.2014.4.1294
[14]	B.-Y. Xi, F. Qi, Some Hermite–Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat., 42 (2013), 243–257.
[15]	B.-Y. Xi, F. Qi, Some integral inequalities of Hermite–Hadamard type for $s$-logarithmically convex functions(Chinese), Acta Math. Sci. Ser. A Chin. Ed, 35 (2015), 515–524.
[16]	B.-Y. Xi, F. Qi, Inequalities of Hermite–Hadamard type for extended $s$-convex functions and applications to means, J. Nonlinear Convex Anal., 16 (2015), 873–890.
[17]	B.-Y. Xi, F. Qi, T.-Y. Zhang, Some inequalities of Hermite–Hadamard type for $m$-harmonic-arithmetically convex functions, ScienceAsia, 41 (2015), 357–361. http://dx.doi.org/10.2306/scienceasia1513-1874.2015.41.357 doi: 10.2306/scienceasia1513-1874.2015.41.357
[18]	B.-Y. Xi, Y. Wang, F. Qi, Some integral inequalities of Hermite–Hadamard type for extended $(s, m)$-convex functions, Transylv. J. Math. Mech., 5 (2013), 69–84.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)