Original Research Article

Article volume = 2021 and issue = 1

Pages: 33–43

Article publication Date: November, 1, 2021

You can download PDF file of the article here: Download

Visited 95 times and downloaded 101 times

Inequalities Of the Hermite-Hadamard Type, for Functions (H, M)-Convex Modified of the Second Type

Juan E. N´apoles V.(a), Florencia Rabossi(b), Hijaz Ahmad(c)

(a)UNNE, FaCENA Ave. Libertad 5450, Corrientes 3400, Argentina and UTN-FRRE, French 414, Resistencia, Chaco 3500, Argentina

(b)UNNE, FaCENA, Ave. Libertad 5450, Corrientes 3400, Argentina

(c)University of Engineering and Technology, Peshawar, Pakistan and International Telematic University Uninettuno, Italy


Abstract:

In this paper we obtain some extensions of the Hermite-Hadamard Inequality for functions (h,m)-convex modified of second type, using the framework of weighted integrals. We show throughout the work that several known results are particular cases of ours.

Keywords:

Hermite-Hadamard integral inequality, integral operators weighted, (h,m)-convex modified functions.


References:
  • [1] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for (k, h)-Riemann-Liouville fractional integral, NTMSCI 4, No. 1, 138-146 (2016) http://dx.doi.org/10.20852/ntmsci.2016217824 4
  • [2] M. A. Ali, J. E. N´apoles V., A. Kashuri, Z. Zhang, Fractional non conformable Hermite-Hadamard inequalities for generalized -convex functions, Fasciculi Mathematici, Nr 64 2020, 5-16 DOI: 10.21008/j.0044-4413.2020.0007. 1
  • [3] M. Alomari, M. Darus, S.S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex. Tamkang J. Math. 41(4), 353-359 (2010) 2.2
  • [4] B. Bayraktar, Some New Inequalities of Hermite-Hadamard Type for Differentiable Godunova-Levin Functions via Fractional Integrals, Konuralp Journal of Mathematics, 8 (1) (2020) 91-96 2.2
  • [5] B. Bayraktar, SOME INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DIFFERENTIABLE (s,m)-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS, TWMS J. App. Eng. Math. V.10, N.3, 2020, pp. 625-637 2.3, 2.6, 2.9, 2.12
  • [6] B. Bayraktar, J. E. N´apoles V., A note on Hermite-Hadamard integral inequality for (h,m)-convex modified functions in a generalized framework, submited. 1
  • [7] S. Bermudo, P. K´orus, J. E. N´apoles V., On q-Hermite-Hadamard inequalities for general convex functions, Acta Math. Hungar. 162, 364-374 (2020) https://doi.org/10.1007/s10474-020-01025-6 1
  • [8] M. Bessenyei, Z. P´ales, On generalized higher-order convexity and Hermite–Hadamard-type inequalities, Acta Sci. Math. (Szeged), 70 (2004), no. 1-2, 13-24. 1
  • [9] M. Bilal, M. Imtiaz, A. R. Khan, I. U. Khan, M. Zafran, Generalized Hermite-Hadamard inequalities for s-convex functions in the mixed kind, submitted 9
  • [10] W. W. Breckner, Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen, Pupl. Inst. Math., 23 (1978), 13-20. 6, 1
  • [11] A. M. Bruckner, E. Ostrow, SOME FUNCTION CLASSES RELATED TO THE CLASS OF CONVEX FUNCTIONS, Pacific J. Math. 12 (1962), 1203-1215 1, 2, 3, 4
  • [12] F. X. Chen, ON THE GENERALIZATION OF SOME HERMITE-HADAMARD INEQUALITIES FOR FUNCTIONS WITH CONVEX ABSOLUTE VALUES OF THE SECOND DERIVATIVES VIA FRACTIONAL INTEGRALS, Ukrainian Mathematical Journal, Vol. 70, No. 12, May, 2019 DOI 10.1007/s11253-019-01618-7 2.3
  • [13] R. D´ıaz, E. Pariguan, On hypergeometric functions and Pochhammerk-symbol. Divulg. Mat. 15(2), 179-192 (2007). 1
  • [14] S. S. Dragomir, S. Fitzpatrik, The Hadamard’s inequality for s -convex functions in the second sense, Demonstration Math., 32 (4) (1999), 687-696. 1
  • [15] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities, RGMIA Monographs, Victoria University, 2000, available at http://rgmia.vu.edu.au/monographs/hermite hadamard.html. 1
  • [16] S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-241. 1
  • [17] P. M. Guzm´an, J. E. N´apoles V., Y. Gasimov, Integral inequalities within the framework of generalized fractional integrals, Fractional Differential Calculus, Volume 11, Number 1 (2021), 69-84 doi:10.7153/fdc-2021-11-05 1
  • [18] J. Hadamard, ´ Etude sur les propri´et´es des fonctions enti´eres et en particulier d’une fonction consid´er´ee par Riemann, J. Math. Pures App. 9, 171-216 (1893). 1
  • [19] C. Hermite, Sur deux limites d’une int´egrale d´efinie, Mathesis3, 82 (1883). 1
  • [20] J. E. Hern´andez Hern´andez, On Some New Integral Inequalities Related With The Hermite-Hadamard Inequality via h-Convex Functions, MAYFEB Journal of Mathematics, Vol 4 (2017), 1-12 1
  • [21] H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994), no. 1, 100-111 6
  • [22] R. Hussain, A. Ali, G. Gulshani, A. Latif, K. Rauf, HERMITE-HADAMARD TYPE INEQUALITIES FOR k- RIEMANN LIOUVILLE FRACTIONAL INTEGRALS VIA TWO KINDS OF CONVEXITY, AJMAA, Vol. 13, No. 1, Art. 17, 1-12, 2016. 2.2, 2.9, 2.12
  • [23] F. Jarad, T. Abdeljawad, T. Shah, ON THE WEIGHTED FRACTIONAL OPERATORS OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION, Fractals, Vol. 28, No. 8 (2020) 2040011 (12 pages) DOI: 10.1142/S0218348X20400113 5
  • [24] F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ. 2017, 2017, 247. 7
  • [25] T. U. Khan, M. A. Khan, Generalized conformable fractional integral operators, J. Comput. Appl. Math. 2019, 346, 378-389. 7
  • [26] S. Kermausuor, E. R. Nwaeze, A. M. Tameru, New Integral Inequalities via the Katugampola Fractional Integrals for Functions Whose Second Derivatives Are Strongly -Convex, Mathematics 2019, 7, 183; doi:10.3390/math7020183 2.3
  • [27] M. Klariˇci´c, E. Neuman, J. Peˇcari´c, V. ˇSimi´c, Hermite–Hadamard’s inequalities for multivariate g-convex functions, Math. Inequal. Appl., 8 (2005), no. 2, 305-316. 1
  • [28] M. Matloka, ON SOME INTEGRAL INEQUALITIES FOR (h,m)-CONVEX FUNCTIONS, MATHEMATICAL ECONOMICS No. 9(16) 2013, 55-70 1
  • [29] V. G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj- Napoca (Romania) (1993). 10
  • [30] P. O. Mohammed, Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions, Turkish Journal of Analysis and Number Theory, 2016, Vol. 4, No. 5, 135-139 2.3
  • [31] M. S. Moslehian, Matrix Hermite–Hadamard type inequalities, Houston J. Math., 39 (2013), No. 1, 177-189. 1
  • [32] S. Mubeen, G. M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2), 2012, 89-94. 3
  • [33] M. A. Noor, K. I. Noor, M. U. Awan, GENERALIZED FRACTIONAL HERMITE-HADAMARD INEQUALITIES, Miskolc Mathematical Notes, Vol. 21 (2020), No. 2, 1001-1011 DOI: 10.18514/MMN.2020.1143 2.2, 2.9, 2.12
  • [34] J. E. N´apoles Vald´es, F. Rabossi, A. D. Samaniego, CONVEX FUNCTIONS: ARIADNE’S THREAD OR CHARLOTTE’S SPIDERWEB?, Advanced Mathematical Models & Applications Vol.5, No.2, 2020, pp.176-191 1
  • [35] J. E. N´apoles Vald´es, J. M. Rodr´ıguez, J. M. Sigarreta, On Hermite-Hadamard type inequalities for nonconformable integral operators, Symmetry 2019, 11, 1108. 1
  • [36] M. A. Noor, G. Cristescu, M. U. Awan, Generalized Fractional Hermite-Hadamard Inequalities for Twice Differentiable s-Convex Functions, Filomat 29:4 (2015), 807-815 DOI 10.2298/FIL1504807N 2.2, 2.5, 2.8, 2.11
  • [37] M. E. O¨ zdemir, A. O. Akdemri, E. Set, On (h,m)-convexity and Hadamard-type inequalities, Transylv. J. Math.Mech. 8(1), 51-58 (2016) 12
  • [38] M. E. O¨ zdemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via ( ,m)-convexity, Computers and Mathematics with Applications 61 (2011) 2614-2620 2.9, 2.12
  • [39] M. E. O¨ zdemir, M. Avci, E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Applied Mathematics Letters 23 (2010) 1065-1070 2.12
  • [40] M. E. O¨ zdemir, S. I. Butt, B. Bayraktar, J. Nasir, Several integral inequalities for (a,s,m)-convex functions, AIMS Mathematics, 5(4): 3906-3921. DOI: 10.3934/math.2020253 2.9, 2.12
  • [41] M. E. O¨ zdemir, A. Ekinci, A. O. Akdemir, GENERALIZATIONS OF INTEGRAL INEQUALITIES FOR FUNCTIONS WHOSE SECOND DERIVATIVES ARE CONVEX AND m-CONVEX, Miskolc Mathematical Notes, Vol. 13 (2012), No. 2, pp. 441-457 2.3, 2.6, 2.9, 2.12
  • [42] M. E. O¨ zdemir, M. Gurbuz, C. Yildiz, INEQUALITIES FOR MAPPINGS WHOSE SECOND DERIVATIVES ARE QUASI-CONVEX OR h-CONVEX FUNCTIONS, Miskolc Mathematical Notes, Vol. 15 (2014), No. 2, pp. 635–649 2.6, 2.12
  • [43] M. E. O¨ zdemir, C. Yildiz, A. O. Akdemir, E. Set, On some inequalities for s-convex functions and applications, Journal of Inequalities and Applications 2013, 2013:333 2.6, 2.9, 2.12
  • [44] J. Park, Generalization of Ostrowski-type inequalities for differentiable real (s,m)-convex mappings, Far East J. Math. Sci., 49 (2011), 157-171. 8
  • [45] F. Qi, B. N. Guo, Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. Rev. R. Acad. Cienc. Exactas F´ıs. Nat., Ser. A Mat. 111(2), 425-434 (2017). https://doi.org/10.1007/s13398-016-0302-6 1
  • [46] E. D. Rainville, Special Functions. Macmillan Co., New York (1960). 1
  • [47] M. Z. Sarikaya, F. Ertugral, On the generalized Hermite-Hadamard inequalities, Annals of the University of Craiova, Mathematics and Computer Science Series Volume 47(1), 2020, Pages 193-213 6
  • [48] M. Z. Sarikaya, A. Saglam, H. Yildirim, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, International Journal of Open Problems in Computer Science and Mathematics 5(3) 2012 DOI: 10.12816/0006114 1, 2.2, 2.6, 2.9, 2.12
  • [49] E. Set, A. G¨ozpinar, J. Choi, HERMITE-HADAMARD TYPE INEQUALITIES FOR TWICE DIFFERENTIABLE m-CONVEX FUNCTIONS VIA CONFORMABLE FRACTIONAL INTEGRALS, Far East Journal of Mathematical Sciences (FJMS), Volume 101, Number 4, 2017, Pages 873-891 http://dx.doi.org/10.17654/MS101040873 2.2, 2.3
  • [50] G. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, University Cluj-Napoca, 1985, 329-338. 5
  • [51] J. R. Wang, X. Li, M. Feckan, Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Applicable Analysis 2012, 1-13, iFirst DOI: 10.1080/00036811.2012.727986. 2.2, 2.9, 2.12
  • [52] 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), no. 5, 873-890. 7
  • [53] B. Y. Xi, D.D. Gao, F. Qi, Integral inequalities of Hermite-Hadamard type for ( , s)-convex and ( , s,m)-convex functions, ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N.44-2020 (499-510) 11, 2.2, 2.9, 2.12
  • [54] Z. H. Yang, J. F. Tian, Monotonicity and inequalities for the gamma function. J. Inequal. Appl. 2017, 317 (2017). https://doi.org/10.1186/s13660-017-1591-9 1
  • [55] Z. H. Yang, J. F. Tian, Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1ˆa22 (2018). https://doi.org/10.7153/jmi-2018-12-01 1
Cite this article as:
  • Juan E. N´apoles V., Florencia Rabossi, Hijaz Ahmad, Inequalities Of the Hermite-Hadamard Type, for Functions (H, M)-Convex Modified of the Second Type, Communications in Combinatorics, Cryptography & Computer Science, 2021(1), PP.33–43, 2021
  • Export citation to BibTeX