On GT-convexity and related integral inequalities

Shu-Hong Wang; Xiao-Wei Sun; Bai-Ni Guo; Shu-Hong Wang; Xiao-Wei Sun; Bai-Ni Guo

doi:10.3934/math.2020255

AIMS Mathematics

2020, Volume 5, Issue 4: 3952-3965. doi: 10.3934/math.2020255

Previous Article Next Article

Research article

On GT-convexity and related integral inequalities

1.
College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
2.
Tongliao City Water Supply Co. LTD, Tongliao 028000, Inner Mongolia, China
3.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

Received: 23 March 2020 Accepted: 21 April 2020 Published: 26 April 2020
MSC : 26A51, 26B12, 41A55

In the paper, the authors introduce a new class of convex functions, GT-convex functions, establish some integral inequalities for GT-convex functions and for the product of two GT-convex functions, and give some applications to classical special means.
- GT-convexity,
- MT-convexity,
- GA-convexity,
- integral inequality,
- mean inequality
Citation: Shu-Hong Wang, Xiao-Wei Sun, Bai-Ni Guo. On GT-convexity and related integral inequalities[J]. AIMS Mathematics, 2020, 5(4): 3952-3965. doi: 10.3934/math.2020255

Related Papers:

Abstract

In the paper, the authors introduce a new class of convex functions, GT-convex functions, establish some integral inequalities for GT-convex functions and for the product of two GT-convex functions, and give some applications to classical special means.

References

[1]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 335 (2007), 1294-1308. doi: 10.1016/j.jmaa.2007.02.016
[2]	Y. M. Chu, M. A. Khan, T. U. Khan, et al. Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305-4316. doi: 10.22436/jnsa.009.06.72
[3]	W. Liu, Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Math. Notes, 16 (2015), 249-256. doi: 10.18514/MMN.2015.1131
[4]	W. Liu, W. Wen, J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9 (2016), 766-777. doi: 10.22436/jnsa.009.03.05
[5]	M. A. Noor, K. I. Noor, M. U. Awan, Some inequalities for geometrically-arithmetically h-convex functions, Creat. Math. Inform., 23 (2014), 91-98. doi: 10.37193/CMI.2014.01.14
[6]	J. Park, Hermite-Hadamard-like type inequalities for twice differentiable MT-convex functions, Appl. Math. Sci., 9 (2015), 5235-5250.
[7]	M. Tunç, Y. Subas, I. Karabayir, On some Hadamard type inequalities for MT-convex functions, Int. J. Open Problems Compt. Math., 6 (2013), 101-113.
[8]	S. H. Wang, Hermite-Hadamard type inequalities for operator convex functions on the coordinates, J. Nonlinear Sci. Appl., 10 (2017), 1116-1125. doi: 10.22436/jnsa.010.03.22
[9]	S. H. Wang, F. Qi, Inequalities of Hermite-Hadamard type for convex functions which are n-times differentiable, Math. Inequal. Appl., 16 (2013), 1269-1278.
[10]	B. Y. Xi, F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Sci. Math. Hungar., 51 (2014), 530-546.
[11]	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.
[12]	X. M. Zhang, Y. M. Chu, X. H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its application, J. Inequal. Appl., 2010, Art. ID 507560, 11 pages, Available from: https://doi.org/10.1155/2010/507560.
[13]	T. H. Zhao, Y. M. Chu, Y. P. Jiang, Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl., 2009, Art. ID 728612, 13 pages, Available from: http://dx.doi.org/10.1155/2009/728612.
[14]	B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6 (2003), Suppl., Art. 1, Available from: http://rgmia.org/v6(E).php.
[15]	Y. Wu, F. Qi, D. W. Niu, Integral inequalities of Hermite-Hadamard type for the product of strongly logarithmically convex and other convex functions, Maejo Int. J. Sci. Technol., 9 (2015), 394-402.
[16]	H. P. Yin, F. Qi, Hermite-Hadamard type inequalities for the product of (α, m)-convex functions, J. Nonlinear Sci. Appl., 8 (2015), 231-236. doi: 10.22436/jnsa.008.03.07
[17]	H. P. Yin, F. Qi, Hermite-Hadamard type inequalities for the product of (α, m)-convex functions, Missouri J. Math. Sci., 27 (2015), 71-79. doi: 10.35834/mjms/1449161369
[18]	J. Q. Wang, B. N. Guo, F. Qi, Generalizations and applications of Young's integral inequality by higher order derivatives, J. Inequal. Appl., 2019, Paper No. 243, 18 pages, Available from: https://doi.org/10.1186/s13660-019-2196-2.
[19]	M. Tunç, H. Yildirim, On MT-convexity, arXiv preprint (2012), Available from: https://arxiv.org/abs/1205.5453.
[20]	C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2000), 155-167.
[21]	C. P. Niculescu, Convexity according to means, Math. Inequal. Appl., 6 (2003), 571-579.
[22]	T. Y. Zhang, A. P. Ji, F. Qi, Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means, Matematiche (Catania), 68 (2013), 229-239.
[23]	S. S. Dragomir., Inequalities of Hermite-Hadamard type for GA-convex functions, Ann. Math. Sil., 32 (2018), 145-168.
[24]	H. J. Skala, On the characterization of certain similarly ordered super-additive functionals, Proc. Amer. Math. Soc., 126 (1998), 1349-1353. doi: 10.1090/S0002-9939-98-04702-9
[25]	B. Y. Xi, D. D. Gao, T. Zhang, et al. Shannon type inequalities for Kapur's entropy, Mathematics, 7 (2019), no. 1, Article 22, 8 pages, Available from: https://doi.org/10.3390/math7010022.
[26]	B. Y. Xi, Y. Wu, H. N. Shi, et al. Generalizations of several inequalities related to multivariate geometric means, Mathematics, 7 (2019), no. 6, Article 552, 15 pages, Available from: https://doi.org/10.3390/math7060552.
[27]	I. A. Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Spaces, 2020, Art. ID 3075390, 7 pages; Available from: https://doi.org/10.1155/2020/3075390.
[28]	M. A. Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Difference Equ., 2020, Paper No. 99, 20 pages, Available from: https://doi.org/10.1186/s13662-020-02559-3.
[29]	A. Iqbal, M. A. Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Spaces, 2020, Art. ID 9845407, 18 pages, Available from: https://doi.org/10.1155/2020/9845407.
[30]	M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for coordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019, Paper No. 317, 33 pages, Available from: https://doi.org/10.1186/s13660-019-2272-7.
[31]	T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), no. 2, Article ID 96, 14 pages, Available from: https://doi.org/10.1007/s13398-020-00825-3.

Reader Comments

Your name:*

Email:*
© 2020 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)