Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets
Abstract - 293
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Keywords

Arithmetic mean
Geometric mean
Jensen inequality
Convex fuzzy sets

How to Cite

Singh, D., & Singh, J. N. (2023). Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets. Journal of Advances in Applied & Computational Mathematics, 10, 71–76. https://doi.org/10.15377/2409-5761.2023.10.7

Abstract

Convex analysis is a discipline of mathematics dedicated to the explication of the properties of convex sets and convex functions. Convex functions are extremely useful in proving many famous inequalities in mathematics. It is widely known that inequalities defined by convex functions originated with the works done by Holder and Jensen among others and applied to modeling a variety of problems both in hard and soft sciences. It is said that the arsenal of an analyst is heavily stocked with inequalities. Studies related to convexity have kept occupying a central position in almost all areas of mathematics, especially in functional analysis and operations research. Following the emergence of fuzzy set theory, fuzzy convexity alongside a number of related concepts has been explicated. However, not much has been done regarding fuzzy inequalities defined by fuzzy convexity. In this paper, a novel attempt to study arithmetic mean-geometric mean is proposed. In this short note, we provide two proofs of the arithmetic mean-geometric mean inequality for convex fuzzy sets.

Mathematics Subject Classification 2020: 03E72, and 94D05.

https://doi.org/10.15377/2409-5761.2023.10.7
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References

Avriel M, Diewert WE, Schaible S, Zang I. Generalized concavity. Boston, MA: Plenum Press; 1988. https://doi.org/10.1007/978-1-4684-7600-2

Boyd S, Vandenberghe L. Convex optimization. Cambridge University Press; 2004. https://doi.org/10.1017/CBO9780511804441

Dubois D, Esteva F, Godo L, Prade H. An information-based discussion of vagueness. IEEE International Conference on Fuzzy Systems, vol. 2, Melbourne: 2001, p. 781-4. https://doi.org/10.1109/FUZZ.2001.1009071

Drewniak J. Convex and strongly convex fuzzy sets. J Math Anal Appl. 1987; 126: 292-300. https://doi.org/10.1016/0022-247X(87)90093-X

Foo N, Low BT. A Note on prototypes, convexity and fuzzy sets. Stud Log. 2008; 90: 125-37. https://doi.org/10.1007/s11225-008-9146-1

Gadjiev D, Kochetkov I, Rustanov A. The convex fuzzy sets and their properties with application to the modeling with fuzzy convex membership functions. In: Murgual V, Pukhkal V, editors. Advances in Intelligent Systems and Computing, EMMFT: Springer; 2021, p. 276-84. https://doi.org/10.1007/978-3-030-57453-6_24

Jain R. Tolerance analysis using fuzzy sets. Int J Syst. 1976; 7: 1393-401. https://doi.org/10.1080/00207727608942013

Matłoka M. A Note on convex fuzzy processes. Adv Fuzzy Syst. 2012; 2012: 1-4. https://doi.org/10.1155/2012/290845

Rockafellar RT. Convex analysis. Princeton University Press; 1970. https://doi.org/10.1515/9781400873173

Syau Y-R. Closed and convex fuzzy sets. Fuzzy Sets Syst. 2000; 110: 287-91. https://doi.org/10.1016/S0165-0114(98)00082-7

Tremblay J, Manohar R. Discrete mathematical structures with applications to computer science. McGraw Hill; 1975.

Youness EA. E-Convex Sets, E-Convex Functions, and E-Convex Programming. J Optim Theory Appl. 1999; 102: 439-50. https://doi.org/10.1023/A:1021792726715

Ammar EE. Some properties of convex fuzzy sets and convex fuzzy cones. Fuzzy Sets Syst. 1999; 106: 381-6. https://doi.org/10.1016/S0165-0114(97)00273-X

Ammar E, Metz J. On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets Syst. 1992; 49: 135-41. https://doi.org/10.1016/0165-0114(92)90319-Y

Bellman RE, Zadeh LA. Decision-Making in a Fuzzy Environment. Manage Sci. 1970; 17: 141-64. https://doi.org/10.1287/mnsc.17.4.b141

Chalco-Cano Y, Rojas-Medar MA, Osuna-Gómez R. S-convex fuzzy processes. Comput Math Appl. 2004; 47: 1411-8. https://doi.org/10.1016/S0898-1221(04)90133-2

Dubois D, Prade H. Fuzzy Sets and Systems: Theory and Applications. Academic Press 1980; 33. https://doi.org/10.2307/2581310

Proctor CH, Kaufman A, Swanson DL. Introduction to the theory of fuzzy subsets. vol. 1. NY: 1975. https://doi.org/10.2307/2286947

Klir G, Yuan B. Fuzzy sets and fuzzy logic: Theory and applications. Prentice Hall; 1995.

Yang X. Some properties of convex fuzzy sets. Fuzzy Sets Syst. 1995; 72: 129-32. https://doi.org/10.1016/0165-0114(94)00285-F

Lowen R. Convex fuzzy sets. Fuzzy Sets Syst. 1980; 3: 291-310. https://doi.org/10.1016/0165-0114(80)90025-1

Matłoka M. Convex fuzzy processes. Fuzzy Sets Syst. 2000; 110: 109-14. https://doi.org/10.1016/S0165-0114(98)00053-0

Matłoka M. Some properties of Φ-convex fuzzy processes. J Fuzzy Math. 1999; 7: 706-26. https://doi.org/10.25291/VR/1999-1-VR-706

Syau Y-R, Low C-Y, Wu T-H. A note on convex fuzzy processes. Appl Math Lett. 2002; 15: 193-6. https://doi.org/10.1016/S0893-9659(01)00117-3

Wu S-Y, Cheng W-H. A note on fuzzy convexity. Appl Math Lett. 2004; 17: 1127-33. https://doi.org/10.1016/j.aml.2003.11.003

Yang X. Some properties of convex fuzzy sets. Fuzzy Sets Syst. 1995; 72: 129-32. https://doi.org/10.1016/0165-0114(94)00285-F

Yatabe S, Inaoka H. On evans’s vague object from set theoretic viewpoint. J Philos Logic. 2006; 35: 423-34. https://doi.org/10.1007/s10992-005-9022-7

Hölder O. Ueber einen Mittelwertsatz Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen. Band (in German) 1889; (2): 38-47.

Jensen JLWV. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 1906; 30: 175-93. https://doi.org/10.1007/BF02418571

Bollobás B. Linear analysis: An introductory course. 2nd ed. Cambridge University Press; 1999. https://doi.org/10.1017/CBO9781139168472

Zadeh LA. Fuzzy sets. Inf Control. 1965; 8: 338-53. https://doi.org/10.1016/S0019-9958(65)90241-X

Yang X. Convexity of semicontinuous functions. J Oper Res Soc Ind. 1994; 31: 309-17.

Singh D, Singh J. Jensen’s inequality for convex fuzzy sets. J Fuzzy Math. 2019; 27: 743–8.

Cvetkovski Z. Inequalities theorems, techniques, and selected problems. Berlin, Heidelberg: Springer Berlin Heidelberg; 2012. https://doi.org/10.1007/978-3-642-23792-8

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Copyright (c) 2023 Dashrath Singh, Jai N. Singh