On Fejer type inequalities for co-ordinated hyperbolic <em>ρ</em>-convex functions

Hasan Kara; Hüseyin Budak; Mehmet Eyüp Kiriş; Hasan Kara; Hüseyin Budak; Mehmet Eyüp Kiriş

doi:10.3934/math.2020300

AIMS Mathematics

2020, Volume 5, Issue 5: 4681-4701. doi: 10.3934/math.2020300

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Research article

On Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions

1.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
2.
Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkey

Received: 27 November 2019 Accepted: 21 May 2020 Published: 27 May 2020
MSC : 26D07, 26D10, 26D15, 26A33

In this study, we first establish some Hermite-Hadamard-Fejer type inequalities for coordinated hyperbolic ρ-convex functions. Then, by utilizing these inequalities, we also give some fractional Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions. The inequalities obtained in this study provide generalizations of some result given in earlier works.
- convex function,
- hyperbolic ρ-convex functions,
- Fejer inequality,
- fractional integrals
Citation: Hasan Kara, Hüseyin Budak, Mehmet Eyüp Kiriş. On Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions[J]. AIMS Mathematics, 2020, 5(5): 4681-4701. doi: 10.3934/math.2020300

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Abstract

In this study, we first establish some Hermite-Hadamard-Fejer type inequalities for coordinated hyperbolic ρ-convex functions. Then, by utilizing these inequalities, we also give some fractional Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions. The inequalities obtained in this study provide generalizations of some result given in earlier works.

References

[1]	A. Akkurt, M. Z. Sarikaya, H. Budak, et al. On the Hadamard's type inequalities for co-ordinated convex functions via fractional integrals, Journal of King Saud University - Science, 29 (2017), 380-387. doi: 10.1016/j.jksus.2016.06.003
[2]	T. Ali, M. A. Khan, A. Kilicman, et al. On the refined Hermite-Hadamard inequalities, Mathematical Sciences & Applications E-Notes, 6 (2018), 85-92.
[3]	A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Math., 28 (1994), 7-12.
[4]	M. K. Bakula, An improvement of the Hermite-Hadamard inequality for functions convex on the coordinates, Australian journal of mathematical analysis and applications, 11 (2014), 1-7.
[5]	H. Budak and M. Z. Sarikaya, Hermite-Hadamard-Fejer inequalities for double integrals, submitted, 2018.
[6]	F. Chen, A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates, J. Math. Inequal., 8 (2014), 915-923.
[7]	F. Chen, A note on Hermite-Hadamard inequalities for products of convex functions, J. Appl. Math., 2013 (2013), 935020.
[8]	F. Chen, S. Wu, Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 705-716. doi: 10.22436/jnsa.009.02.32
[9]	S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000.
[10]	S. S. Dragomir, Some inequalities of Hermite-Hadamard type for hyperbolic ρ-convex functions, Preprint, 2018.
[11]	S. S. Dragomir, On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan J. Math., 4 (2001), 775-788.
[12]	S. S. Dragomir, Some inequalities of Fejer type for hyperbolic ρ-convex functions, Preprint, 2018.
[13]	S. S. Dragomir, Some inequalities of Ostrowski and trapezoid type for hyperbolic ρ-convex functions, Preprint, 2018.
[14]	S. S. Dragomir, Some inequalities of Jensen type for hyperbolic ρ-convex functions, Preprint, 2018.
[15]	S. S. Dragomir and B. T. Torebek, Some Hermite-Hadamard type inequalities in the class of hyperbolic ρ-convex functions, arXiv preprint, arXiv:1901.06634, 2019.
[16]	G. Farid, M. Marwan and A. U. Rehman, Fejer-Hadamard inequlality for convex functions on the co-ordinates in a rectangle from the plane, International Journal of Analysis and Applications, 10 (2016), 40-47.
[17]	L. Fejer, Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), 369-390. (Hungarian)
[18]	J. Hadamard, Etude sur les proprietes des fonctions entieres en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
[19]	U. S. Kırmacı, M. K. Bakula, M. E. Özdemir, et al. Hadamard-tpye inequalities for s-convex functions, Appl. Math. Comput., 193 (2007), 26-35.
[20]	M. A. Latif and M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinetes, Int. Math. Forum, 4 (2009), 2327-2338.
[21]	M. A. Latif, S. S. Dragomir, E. Momoniat, Generalization of some Inequalities for differentiable co-ordinated convex functions with applications, Moroccan J. Pure Appl. Anal., 2 (2016), 12-32. doi: 10.7603/s40956-016-0002-4
[22]	M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl., 2012 (2012), 28.
[23]	K. Ozcelik, H. Budak, S. S. Dragomir, On Hermite-Hadamard type inequalities for co-ordinated hyperbolic ρ-convex functions, Submitted, 2019.
[24]	M. E. Ozdemir, C. Yildiz and A. O. Akdemir, On the co-ordinated convex functions, Appl. Math. Inf. Sci., 8 (2014), 1085-1091. doi: 10.12785/amis/080318
[25]	B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll, 6 (2003), 1-9.
[26]	Z. Pavic, Improvements of the Hermite-Hadamard inequality, J. Inequal. Appl., 2015 (2015), 1-11. doi: 10.1186/1029-242X-2015-1
[27]	J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
[28]	M. Z. Sarikaya, E. Set, M. E. Ozdemir, et al.New some Hadamard's type inequalities for coordinated convex functions, Tamsui Oxford Journal of Information and Mathematical Sciences, 28 (2012), 137-152.
[29]	M. Z. Sarikaya, On the Hermite-Hadamard-type inequalities for co-ordinated convex functions via fractional integrals, Integ. Transf. Spec. F., 25 (2014), 134-147. doi: 10.1080/10652469.2013.824436
[30]	E. Set, M. E. Özdemir, S. S. Dragomir, On the Hermite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl., 2010 (2010), 148102.
[31]	K. L. Tseng and S. R. Hwang, New Hermite-Hadamard inequalities and their applications, Filomat, 30 (2016), 3667-3680. doi: 10.2298/FIL1614667T
[32]	D. Y. Wang, K. L. Tseng and G. S. Yang, Some Hadamard's inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwan. J. Math., 11 (2007), 63-73. doi: 10.11650/twjm/1500404635
[33]	R. Xiang and F. Chen, On some integral inequalities related to Hermite-Hadamard-Fejér inequalities for coordinated convex functions, Chinese Journal of Mathematics, 2014 (2014), 796132.
[34]	H. Yaldız, M. Z. Sarikaya, Z. Dahmani, On the Hermite-Hadamard-Fejer-type inequalities for coordinated convex functions via fractional integrals, An International Journal of Optimization and Control: Theories & Applications, 7 (2017), 205-215.
[35]	G. S. Yang and K. L. Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239 (1999), 180-187. doi: 10.1006/jmaa.1999.6506
[36]	G. S. Yang and M. C. Hong, A note on Hadamard's inequality, Tamkang J. Math., 28 (1997), 33-37.

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