General Fritz Carlson's Type Inequality for Sugeno Integrals
© X. Wang and C. Bai. 2011
Received: 18 August 2010
Accepted: 20 January 2011
Published: 7 February 2011
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 . The study of inequalities for Sugeno integral was initiated by Román-Flores et al. [2, 3] and then followed by the others [4–11].
Definition 1.1 (see ).
does not hold.
The following properties of the fuzzy integral are well known and can be found in .
Recently, Caballero and Sadarangani  have shown that in general, the Carlson's integral inequality is not valid in the fuzzy context. And they presented a fuzzy version of Fritz Carlson's integral inequality as follows.
In this paper, our purpose is to give a generalization of the above Fritz Carlson's inequality for fuzzy integrals. Moreover, we will give many interesting corollaries of our main results.
2. Main Results
This section provides a generalization of Fritz Carlson's type inequality for Sugeno integrals. Before stating our main results, we need the following lemmas.
Lemma 2.1 (see ).
Corollary 2.2 (see ).
Note that two monotone functions (in the same sense) are comonotone.
From Theorem 2.4, we get the following.
Corollary 2.5 (see ).
Corollary 2.6 (see ).
The following result presents a fuzzy version of generalized Carlson's inequality.
This inequality implies that (2.11) holds
By Theorem 2.7, we have the following.
which implies that (2.17) holds. Similarly, we can obtain (2.17) by of [7, Theorem 2] for the case of .
From Theorem 2.9 and Corollary 2.8, we have the following.
Corollary 2.8 is a generalization of the main result in [8, Theorem 1].
Thus, by Theorem 2.11 we can get the following corollary.
The authors would like to thank the referees for reading this work carefully, providing valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (no. 10771212).
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