BF-Ostrowski Type Inequalities for (A, B, G, D)-convex
Abstract
In this paper, we present the very first time the generalized notion of (A,B, G,D)−convex (concave) function in mixed kind, which is the generalization of functions given in [15], [2], [4], [14], [16] and [3]. We would like to state well-nown Ostrowski inequality via Fuzzy Riemann Integrals for (A,B, G,D)− convex (concave) function in mixed kind. Moreover we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are (A,B,G,D)-convex (concave) functions in mixed kind by using different techniques including H¨older’s inequality [27] and power mean inequality [26]. Also, various established results would be captured as special cases with respect to convexity of function.
Keywords
Full Text:
PDFReferences
M. Alomari, M. Darus, S. S. Dragomir and P. Cerone: Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl. Math. Lett. 23(1) (2010), pp. 1071—1076.
A. Arshad and A. R. Khan: Hermite−Hadamard−Fejer Type Integral Inequality for s − p−Convex Functions of Several Kinds. TJMM. 11(2) (2019), pp. 25–40.
E. F. Beckenbach: Convex functions. Bull. Amer. Math. Soc. 54(1) (1948), pp. 439–460.
W. W. Breckner: Stetigkeitsaussagen Fur Eine Klasse Verallgemeinerter Konvexer Funktionen in Topologischen Linearen Raumen. (German). Publ. Inst. Math. 37(1) (1978), pp. 13–20.
E. Set: New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals. Comput. Math. Appl. 63(1) (2012), pp. 1147–1154.
M. J. V. Cortez, and J. E. Hernandez: Ostrowski and Jensen-type inequalities via (s, m)-convex functions in the second sense. Bol. Soc. Mat. Mex. 26(1) (2020), pp. 287—302.
S. S. Dragomir: A Companion of Ostrowski’s Inequality for Functions of Bounded Variation and Applications. Int. J. Nonlinear Anal. Appl. 5(1) (2014), pp. 89–97.
S. S. Dragomir: A Functional Generalization of Ostrowski Inequality via Montgomery identity. Acta Math. Univ. Comenianae, LXXXIV. 1(1) (2015), pp. 63–78.
S. S. Dragomir: On the Ostrowski’s Integral Inequality for Mappings with Bounded Variation and Applications. Math. Inequal. Appl. 4(1) (2001), pp. 59–66.
S. S. Dragomir: Refinements of the Generalised Trapozoid and Ostrowski Inequalities for Functions of Bounded Variation. Arch. Math. 91(5) (2008), pp. 450–460.
S. S. Dragomir and N. S. Barnett: An Ostrowski Type Inequality for Mappings whose Second Derivatives are Bounded and Applications. J. Indian Math. Soc. (N.S.)
(4) (1999), pp. 237–245.
S. S. Dragomir, P. Cerone, N. S. Barnett and J. Roumeliotis: An Inequality of the Ostrowski Type for Double Integrals and Applications for Cubature Formulae. Tamsui Oxf. J. Math. Sci. 16(1) (2000), pp. 1–16.
S. S. Dragomir, P. Cerone and J. Roumeliotis: A new Generalization of Ostrowski Integral Inequality for Mappings whose Derivatives are Bounded and Applications in Numerical Integration and for Special Means. Appl. Math. Lett. 13(1) (2000), pp. 19–25.
S. S. Dragomir, J. Pečarić and L. Persson: Some inequalities of Hadamard type. Soochow J. Math. 21(3) (1995), pp. 335–341.
A. Ekinci: Klasik E¸sitsizlikler Yoluyla Konveks Fonksiyonlar i¸cin Integral E¸sitsizlikler. Ph.D. Thesis, Thesis ID: 361162 in tez2.yok.gov.tr Atatürk University, (2014).
W. K. Godunova and V. I. Levin: Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions. Numerical Mathematics and Mathematical Physics, (Russian) 166(1) (1985), pp. 138–142.
N. Irshad, A. R. Khan and A. Nazir: Extension of Ostrowki Type Inequality via Moment Generating Function. Adv. Inequal. Appl. 2(1) (2020), pp. 1–15.
N. Irshad, A. R. Khan and M. A. Shaikh: Generalization of Weighted Ostrowski Inequality with Applications in Numerical Integration. Adv. Ineq. Appl. 7(1) (2019), pp. 1–14.
N. Irshad, A. R. Khan and M. A. Shaikh: Generalized Weighted Ostrowski-Gruss Type Inequality with Applications. Global J. Pure Appl. Math. 15(5) (2019), pp. 675–692.
N. Irshad and A. R. Khan: On Weighted Ostrowski Gruss Inequality with Applications. TJMM 10(1) (2018), pp. 15–22.
N. Irshad and A. R. Khan: Generalization of Ostrowski Inequality for Differentiable functions and its applications to numerical quadrature rules. J. Math. Anal 8(1) (2017), pp. 79–102.
M. A. Noor, and M. U. Awan: Some integral inequalities for two kinds of convexities via fractional integrals. TJMM 5(1) (2013), pp. 129 – 136.
A. M. Ostrowski: Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert. Comment. Math. Helv. 10(1) (1938), pp. 226–227.
S. G. Samko, A. A. Kilbas and O. I. Marichev: Fractional Integrals and Derivatives-Theory and Applications Gordon and Breach. Linghorne, PA, (1993).
E. Set: New inequalities of Ostrowski type for mappings whose derivatives are s−convex in the second sense via fractional integrals. Comput Math Appl 63(1) (2012), pp. 1147–1154.
Z. G. Xiao and A. H. Zhang: Mixed power mean inequalities. Research Communication on Inequalities 8(1) (2002), pp. 15—17.
X. Yang: A note on Holder inequality. Appl. Math. Comput. 134(1) (2003), pp. 319–322.
S. Gal: Approximation theory in fuzzy setting, Chapter 13 in Handbook of Analytic Computational Methods in Applied Mathematics (edited by G. Anastassiou). Chapman and Hall, CRC Press, Boca Raton, New York (2000), pp. 617–666.
W. Congxin and M. Ming: On embedding problem of fuzzy number space: Part 1. Fuzzy Sets and Systems 44 (1) (1991), pp.33–38.
W. Congxin and G. Zengtai: On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets and Systems 120 (1) (2001), pp. 523–532.
O. Kaleva: Fuzzy differential equations. Fuzzy Sets and Systems 24(1) (1987), pp. 301–317.
DOI: https://doi.org/10.22190/FUMI210826035H
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)