Wang, Xiaojing; Bai, Chuanzhi General Fritz Carlson’s type inequality for Sugeno integrals. (English) Zbl 1219.26024 J. Inequal. Appl. 2011, Article ID 761430, 9 p. (2011). Fritz Carlson’s inequality for a real function \(f\) claims that \[ \int_0^{\infty} f(x)\,dx \leq \sqrt{\pi} \left ( \int_0^{\infty} f^2(x)\,dx \right ) ^{\frac 14} \left ( \int_0^{\infty} x^2 f^2(x)\,dx \right ) ^{\frac 14}. \]Its generalization for the Sugeno integral and \(\mu\) a Lebesgue measure by Caballero and Sadarangani reads \[ \int_0^1 f(x)\,d\mu \leq \sqrt{2} \left ( \int_0^1 f^2(x)\,d\mu \right ) ^{\frac 14} \left ( \int_0^1 x^2 f^2(x)\,d\mu \right ) ^{\frac 14}. \]The authors show a version of this inequality on a fuzzy measure space in the following form for comonotone functions \(f\) and \(h\), with Sugeno integrals of \(f, g, h\) not exceeding one: \[ \int_A f(x) \,d\mu \leq \frac 1K \left ( \int_A f^p(x)g^p(x) \,d\mu \right ) ^{\frac{1}{p+q}} \left ( \int_A f^q(x) h^q(x) \,d\mu \right ) ^{\frac {1}{p+q}}. \]Corollaries are derived for the cases of monotonicity and convexity of the considered functions. Reviewer: Vladimír Janiš (Banská Bystrica) Cited in 5 Documents MSC: 26E50 Fuzzy real analysis 26D15 Inequalities for sums, series and integrals 28E10 Fuzzy measure theory Keywords:Sugeno integral; Fritz Carlson’s inequality; fuzzy integral PDFBibTeX XMLCite \textit{X. Wang} and \textit{C. Bai}, J. Inequal. Appl. 2011, Article ID 761430, 9 p. (2011; Zbl 1219.26024) Full Text: DOI EuDML References: [1] Sugeno M: Theory of fuzzy integrals and its applications, Ph.D. Dissertation. Tokyo Institute of Technology; 1974. [2] Flores-Franulič A, Román-Flores H: A Chebyshev type inequality for fuzzy integrals.Applied Mathematics and Computation 2007,190(2):1178-1184. 10.1016/j.amc.2007.02.143 · Zbl 1129.26021 · doi:10.1016/j.amc.2007.02.143 [3] Román-Flores H, Flores-Franulič A, Chalco-Cano Y: A Jensen type inequality for fuzzy integrals.Information Sciences 2007,177(15):3192-3201. 10.1016/j.ins.2007.02.006 · Zbl 1127.28013 · doi:10.1016/j.ins.2007.02.006 [4] Mesiar R, Ouyang Y: General Chebyshev type inequalities for Sugeno integrals.Fuzzy Sets and Systems 2009,160(1):58-64. 10.1016/j.fss.2008.04.002 · Zbl 1183.28035 · doi:10.1016/j.fss.2008.04.002 [5] Román-Flores H, Flores-Franulič A, Chalco-Cano Y: A Hardy-type inequality for fuzzy integrals.Applied Mathematics and Computation 2008,204(1):178-183. 10.1016/j.amc.2008.06.027 · Zbl 1168.26317 · doi:10.1016/j.amc.2008.06.027 [6] Agahi H, Mesiar R, Ouyang Y: General Minkowski type inequalities for Sugeno integrals.Fuzzy Sets and Systems 2010,161(5):708-715. 10.1016/j.fss.2009.10.007 · Zbl 1183.28027 · doi:10.1016/j.fss.2009.10.007 [7] Caballero J, Sadarangani K: Hermite-Hadamard inequality for fuzzy integrals.Applied Mathematics and Computation 2009,215(6):2134-2138. 10.1016/j.amc.2009.08.006 · Zbl 1181.26045 · doi:10.1016/j.amc.2009.08.006 [8] Caballero J, Sadarangani K: Fritz Carlson’s inequality for fuzzy integrals.Computers and Mathematics with Applications 2010,59(8):2763-2767. 10.1016/j.camwa.2010.01.045 · Zbl 1193.28015 · doi:10.1016/j.camwa.2010.01.045 [9] Román-Flores H, Flores-Franulič A, Chalco-Cano Y: The fuzzy integral for monotone functions.Applied Mathematics and Computation 2007,185(1):492-498. 10.1016/j.amc.2006.07.066 · Zbl 1116.26024 · doi:10.1016/j.amc.2006.07.066 [10] Román-Flores H, Flores-Franulič A, Chalco-Cano Y: A convolution type inequality for fuzzy integrals.Applied Mathematics and Computation 2008,195(1):94-99. 10.1016/j.amc.2007.04.072 · Zbl 1149.26033 · doi:10.1016/j.amc.2007.04.072 [11] Caballero J, Sadarangani K: A Cauchy-Schwarz type inequality for fuzzy integrals.Nonlinear Analysis. Theory, Methods and Applications. Series A 2010,73(10):3329-3335. 10.1016/j.na.2010.07.013 · Zbl 1202.26027 · doi:10.1016/j.na.2010.07.013 [12] Wang Z, Klir G: Fuzzy Measure Theory. Plenum Press, New York, NY, USA; 1992:x+354. · Zbl 0812.28010 · doi:10.1007/978-1-4757-5303-5 [13] Carlson F: Une ineqalite.Arkiv för Matematik 1934, 25: 1-5. · Zbl 0009.34202 · doi:10.1007/BF02384433 [14] Hardy GH: A note on two inequalities.Journal of the London Mathematical Society 1936, 11: 167-170. 10.1112/jlms/s1-11.3.167 · Zbl 0014.29804 · doi:10.1112/jlms/s1-11.3.167 [15] Ouyang Y, Fang J, Wang L: Fuzzy Chebyshev type inequality.International Journal of Approximate Reasoning 2008,48(3):829-835. 10.1016/j.ijar.2008.01.004 · Zbl 1185.28025 · doi:10.1016/j.ijar.2008.01.004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.