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Báo cáo hóa học: " Research Article General Fritz Carlson’s Type Inequality for Sugeno Integrals"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article General Fritz Carlson’s Type Inequality for Sugeno Integrals | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 761430 9 pages doi 10.1155 2011 761430 Research Article General Fritz Carlson s Type Inequality for Sugeno Integrals Xiaojing Wang and Chuanzhi Bai Department of Mathematics Huaiyin Normal University Huaian Jiangsu 223300 China Correspondence should be addressed to Chuanzhi Bai czbai8@sohu.com Received 18 August 2010 Revised 23 November 2010 Accepted 20 January 2011 Academic Editor Laszlo Losonczi Copyright 2011 X. Wang and C. Bai. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Fritz Carlson s type inequality for fuzzy integrals is studied in a rather general form. The main results of this paper generalize some previous results. 1. Introduction and Preliminaries Recently the study of fuzzy integral inequalities has gained much attention. The most popular method is using the Sugeno integral 1 . The study of inequalities for Sugeno integral was initiated by Roman-Flores et al. 2 3 and then followed by the others 4-11 . Now we introduce some basic notation and properties. For details we refer the reader to 1 12 . Suppose that s is a Ơ-algebra of subsets of X and let s 0 to be a nonnegative extended real-valued set function. We say that is a fuzzy measure if it satisfies 1 0 0 2 E F e s and E c F imply E fF monotonicity 3 En c s El c E2 c imply limn to En UTO 1 En continuity from below 4 En c s E1 D E2 D E1 to imply limnto-SI E ATO 1 En continuity from above . If f is a nonnegative real-valued function defined on X we will denote by Laf x e X f x a f a the a-level of f for a 0 and L0f x e B f x 0 supp f is the support of f. Note that if a p then f p c f a . Let X s p be a fuzzy measure space by F tX we denote the set of all nonnegative p-measurable functions with respect to s. 2 Journal of Inequalities and .