Research article

2D approximately reciprocal ρ-convex functions and associated integral inequalities

  • Received: 02 April 2020 Accepted: 18 May 2020 Published: 27 May 2020
  • MSC : 26A51, 26D10, 26D15

  • The main objective of this article is to introduce the notion of 2D approximately reciprocal ρ-convex functions, show that this class of functions unifies several other unrelated classes of reciprocal convex functions, obtain several new refinements of the Hermite-Hadamard type inequalities involving 2D approximately reciprocal ρ-convex functions, provide some bounds pertaining to the trapezium like inequalities by using partial differentiable 2D approximately reciprocal ρ-convex functions, and discuss the special cases of the obtained results.

    Citation: Muhammad Uzair Awan, Nousheen Akhtar, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu. 2D approximately reciprocal ρ-convex functions and associated integral inequalities[J]. AIMS Mathematics, 2020, 5(5): 4662-4680. doi: 10.3934/math.2020299

    Related Papers:

  • The main objective of this article is to introduce the notion of 2D approximately reciprocal ρ-convex functions, show that this class of functions unifies several other unrelated classes of reciprocal convex functions, obtain several new refinements of the Hermite-Hadamard type inequalities involving 2D approximately reciprocal ρ-convex functions, provide some bounds pertaining to the trapezium like inequalities by using partial differentiable 2D approximately reciprocal ρ-convex functions, and discuss the special cases of the obtained results.


    加载中


    [1] M. K. Wang, H. H. Chu, Y. M. Li, et al. Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind, Appl. Anal. Discrete Math., 14 (2020), 255-271.
    [2] 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, RACSAM, 114 (2020), 1-14. doi: 10.1007/s13398-019-00732-2
    [3] X. M. Hu, J. F. Tian, Y. M. Chu, et al. On Cauchy-Schwarz inequality for N-tuple diamond-alpha integral, J. Inequal. Appl., 2020 (2020), 1-15. doi: 10.1186/s13660-019-2265-6
    [4] S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions, AIMS Mathematics, 5 (2020), 2629-2645. doi: 10.3934/math.2020171
    [5] S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10. doi: 10.1186/s13660-019-1955-4
    [6] S. Khan, M. Adil Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Method. Appl. Sci., 43 (2020), 2577-2587. doi: 10.1002/mma.6066
    [7] W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 1-12. doi: 10.1007/s13398-019-00732-2
    [8] S. Khan, M. Adil Khan, Y. M. Chu, New converses of Jensen inequality via Green functions with applications, RACSAM, 114 (2020), 114.
    [9] Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.
    [10] Y. M. Chu, M. K. Wang, S. L. Qiu, et al. Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 63 (2012), 1177-1184. doi: 10.1016/j.camwa.2011.12.038
    [11] W. M. Qian, Y. Y. Yang, H. W. Zhang, et al. Optimal two-parameter geometric and arithmetic mean bounds for the Sándor-Yang mean, J. Inequal. Appl., 2019 (2019), 1-12. doi: 10.1186/s13660-019-1955-4
    [12] H. Z. Xu, Y. M. Chu, W. M. Qian, Sharp bounds for the Sándor-Yang means in terms of arithmetic and contra-harmonic means, J. Inequal. Appl., 2018 (2018), 1-13. doi: 10.1186/s13660-017-1594-6
    [13] W. M. Qian, Z. Y. He, H. W. Zhang, et al. Sharp bounds for Neuman means in terms of twoparameter contraharmonic and arithmetic mean, J. Inequal. Appl., 2019 (2019), 1-13. doi: 10.1186/s13660-019-1955-4
    [14] W. M. Qian, W. Zhang, Y. M. Chu, Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes, 20 (2019), 1157-1166.
    [15] M. K. Wang, Z. Y. He, Y. M. Chu, Sharp power mean inequalities for the generalized elliptic integral of the first kind, Comput. Meth. Funct. Th., 20 (2020), 111-124. doi: 10.1007/s40315-020-00298-w
    [16] B. Wang, C. L. Luo, S. H. Li, et al. Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, RACSAM, 114 (2020), 1-10. doi: 10.1007/s13398-019-00732-2
    [17] M. K. Wang, M. Y. Hong, Y. F. Xu, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1-21.
    [18] S. Rashid, F. Jarad, M. A. Noor, et al. Inequalities by means of generalized proportional fractional integral operators with respect another function, Mathematics, 7 (2019), 1-18.
    [19] S. Rashid, F. Jarad, Y. M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Probl. Eng., 2020 (2020), 1-12.
    [20] S. Rashid, F. Jarad, H. Kalsoom, et al. On Pólya-Szegö and Ćebyšev type inequalities via generalized k-fractional integrals, Adv. Differ. Equ., 2020 (2020), 125.
    [21] I. Abbas Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Space., 2020 (2020), 1-17.
    [22] M. Adil Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14. doi: 10.1186/s13660-019-1955-4
    [23] S. Zaheer Ullah, M. Adil Khan, Z. A. Khan, et al. Integral majorization type inequalities for the functions in the sense of strong convexity, J. Funct. Space., 2019 (2019), 1-11.
    [24] S. Rafeeq, H. Kalsoom, S. Hussain, et al. Delay dynamic double integral inequalities on time scales with applications, Adv. Differ. Equ., 2020 (2020), 1-32. doi: 10.1186/s13662-019-2438-0
    [25] S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mappings with application, AIMS Mathematics, 5 (2020), 3525-3546. doi: 10.3934/math.2020229
    [26] İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
    [27] M. A. Noor, K. I. Noor, M. U. Awan, et al. Some integral inequalities for harmonically h-convex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (2015), 5-16.
    [28] M. A. Noor, K. I. Noor, M. U. Awan, Integral inequalities for coordinated harmonically convex functions, Complex Var. Elliptic Equ., 60 (2015), 776-786. doi: 10.1080/17476933.2014.976814
    [29] M. U. Awan, M. A. Noor, M. V. Mihai, et al. On approximately harmonic h-convex functions depending on a given function, Filomat, 33 (2019), 3783-3793. doi: 10.2298/FIL1912783A
    [30] M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33. doi: 10.1186/s13660-019-1955-4
    [31] M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
    [32] A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.
    [33] S. Rashid, M. A. Noor, K. I. Noor, et al. Hermite-Hadamrad type inequalities for the class of convex functions on time scale, Mathematics, 7 (2019), 1-20.
    [34] M. U. Awan, S. Talib, Y. M. Chu, et al. Some new refinements of Hermite-Hadamard-type inequalities involving Ψk-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 1-10.
    [35] M. U. Awan, N. Akhtar, S. Iftikhar, et al. Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12. doi: 10.1186/s13660-019-2265-6
    [36] 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. doi: 10.11650/twjm/1500574995
    [37] H. Budak, H. Kara, M. E. Kiri, On Hermite-Hadamard type inequalities for co-ordinated trigonometrically ρ-convex functions, Tbilisi Math. J., 13 (2020), 1-26. doi: 10.32513/tbilisi/1585015215
    [38] T. H. Zhao, Y. M. Chu, H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011 (2011), 1-13.
    [39] G. J. Hai, T. H. Zhao, Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function, J. Inequal. Appl., 2020 (2020), 1-17. doi: 10.1186/s13660-019-2265-6
    [40] S. L. Qiu, X. Y. Ma, Y. M. Chu, Sharp Landen transformation inequalities for hypergeometric functions, with applications, J. Math. Anal. Appl., 474 (2019), 1306-1337. doi: 10.1016/j.jmaa.2019.02.018
    [41] M. K. Wang, Y. M. Chu, Y. P. Jiang, Ramanujan's cubic transformation inequalities for zerobalanced hypergeometric functions, Rocky Mountain J. Math., 46 (2016), 679-691. doi: 10.1216/RMJ-2016-46-2-679
    [42] M. K. Wang, Y. M. Chu, W. Zhang, Monotonicity and inequalities involving zero-balanced hypergeometric function, Math. Inequal. Appl., 22 (2019), 601-617.
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3396) PDF downloads(239) Cited by(28)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog