## Reversed version of a generalized sharp Hölder’s inequality and its applications.(English)Zbl 1266.94012

Author’s abstract: We give a reversed version of a generalized sharp Hölder inequality which is due to Wu. The results are then used to improve Beckenbach’s inequality and Minkowski’s inequality. Moreover, as an application in information theory, we present a refinement of Singh’s inequality which is a generalized Shannon inequality.

### MSC:

 94A15 Information theory (general) 26D15 Inequalities for sums, series and integrals
Full Text:

### References:

 [1] Agahi, H.; Mesiar, R.; Ouyang, Y., General Minkowski type inequalities for sugeno integrals, Fuzzy sets and systems, 161, 708-715, (2010) · Zbl 1183.28027 [2] Agahi, H.; Mesiar, R.; Ouyang, Y., New general extensions of Chebyshev type inequalities for sugeno integrals, International journal of approximate reasoning, 51, 135-140, (2009) · Zbl 1196.28026 [3] Agahi, H.; Mesiar, R.; Ouyang, Y., Further developments of Chebyshev type inequalities for sugeno integral integrals and T-(S-)evaluators, Kybernetika, 46, 83-95, (2009) · Zbl 1188.28014 [4] Agahi, H.; Román-Flores, H.; Flores-Franulic˘, A., General barnes – godunova – levin type inequalities for sugeno integrals, Information sciences, 181, 6, 1072-1079, (2011) · Zbl 1210.28021 [5] Agahi, H.; Yaghoobi, M.A., A Minkowski type inequality for fuzzy integrals, Journal of uncertain systems, 4, 187-194, (2010) [6] Aldaz, J.M., A stability version of Hölder’s inequality, Journal of mathematical analysis and applications, 343, 842-845, (2008) · Zbl 1138.26308 [7] Beckenbach, E.F., A class of Mean-value functions, The American mathematical monthly, 57, 1-6, (1950) · Zbl 0035.15704 [8] Beckenbach, E.F.; Bellman, R., Inequalities, (1983), Springer-Verlag Berlin · Zbl 0513.26003 [9] Bourin, J.-C.; Lee, E.-Y.; Fujii, M.; Seo, Y., A matrix reverse Hölder inequality, Linear algebra and its applications, 431, 2154-2159, (2009) · Zbl 1179.15021 [10] Chen, T.-Y.; Li, C.-H., Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis, Information sciences, 180, 21, 4207-4222, (2010) [11] Chen, B.; Zhu, Y.; Hu, J.; Príncipe, J.C., Delta-entropy: definition, properties and applications in system identification with quantized data, Information sciences, 181, 7, 1384-1402, (2011) · Zbl 1227.93125 [12] Dehmer, M.; Mowshowitz, A., A history of graph entropy measures, Information sciences, 181, 1, 57-78, (2011) · Zbl 1204.94050 [13] Flores-Franulič, A.; Román-Flores, H., A Chebyshev type inequality for fuzzy integrals, Applied mathematics and computation, 190, 1178-1184, (2007) · Zbl 1129.26021 [14] Fujii, M.; Lee, E.-Y.; Seo, Y., A difference counterpart to a matrix Hölder inequality, Linear algebra and its applications, 432, 10, 2565-2571, (2010) · Zbl 1189.15028 [15] Gavalec, M.; Zimmermann, K., Duality of optimization problems with generalized fuzzy relation equation and inequality constraints, Information sciences, (2011) [16] Golshani, L.; Pasha, E., Rényi entropy rate for Gaussian processes, Information sciences, 180, 8, 1486-1491, (2010) · Zbl 1185.60034 [17] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge University Press UK · Zbl 0047.05302 [18] Haber, R.E.; Del Toro, R.M.; Gajate, A., Optimal fuzzy control system using the cross-entropy method. A case study of a drilling process, Information sciences, 180, 14, 2777-2792, (2010) [19] Hu, K., On an inequality and its applications, Scientia sinica, 24, 8, 1047-1055, (1981) · Zbl 0471.26009 [20] Khrennikov, A., Nonlocality as well as rejection of realism are only sufficient(but non-necessary!)conditions for violation of bell’s inequality, Information sciences, 179, 492-504, (2009) · Zbl 1165.81302 [21] Kuang, J., Applied inequalities, (2010), Shandong Science and Technology Press Jinan [22] Kwon, E.G.; Bae, E.K., On a continuous form of Hölder inequality, Journal of mathematical analysis and applications, 343, 1, 585-592, (2008) · Zbl 1138.26314 [23] Li, Y.; Wu, H., Global stability analysis in cohen – grossberg neural networks with delays and inverse Hölder neuron activation functions, Information sciences, 180, 20, 4022-4030, (2010) · Zbl 1195.93124 [24] Lu, J.; Wu, K.; Lin, J., Fast full search in motion estimation by hierarchical use of minkowski’s inequality, Pattern recognition, 31, 945-952, (1998) [25] Małyszko, D.; Stepaniuk, J., Adaptive multilevel rough entropy evolutionary thresholding, Information sciences, 180, 7, 1138-1158, (2010) [26] Mesiar, R.; Mesiarová, A., Fuzzy integrals and linearity, International journal of approximate reasoning, 47, 352-358, (2008) · Zbl 1183.28034 [27] Mesiar, R.; Ouyang, Y., General Chebyshev type inequalities for sugeno integrals, Fuzzy sets and systems, 160, 58-64, (2009) · Zbl 1183.28035 [28] Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M., Classical and new inequalities in analysis, (1993), Kluwer Academic Dordrecht · Zbl 0771.26009 [29] Mond, B.; Pečarić, J.E., On converses of Hölder and Beckenbach inequalities, Journal of mathematical analysis and applications, 196, 795-799, (1995) · Zbl 0857.26007 [30] Ouyang, Y.; Mesiar, R., On the Chebyshev type inequality for seminormed fuzzy integral, Applied mathematics letters, 22, 1810-1815, (2009) · Zbl 1185.28026 [31] Ouyang, Y.; Mesiar, R., Sugeno integral and the comonotone commuting property, International journal of uncertainty, fuzziness and knowledge-based systems, 17, 465-480, (2009) · Zbl 1178.28031 [32] Ouyang, Y.; Mesiar, R.; Agahi, H., An inequality related to Minkowski type for sugeno integrals, Information sciences, 180, 2793-2801, (2010) · Zbl 1193.28016 [33] Ouyang, Y.; Mesiar, R.; Li, J., On the comonotonic-&z.star;-property for sugeno integral, Applied mathematics and computation, 211, 450-458, (2009) · Zbl 1175.28011 [34] Özkan, U.M.; Sarikaya, M.Z.; Yildirim, H., Extensions of certain integral inequalities on time scales, Applied mathematics letters, 21, 993-1000, (2008) · Zbl 1168.26316 [35] Pap, E.; Štrboja, M., Generalization of the Jensen inequality for pseudo-integral, Information sciences, 180, 543-548, (2010) · Zbl 1183.26039 [36] A. Renyi, On measures of entropy and information, in: Proceedings of the Fourth Berkeley Symposium Mathematical Statistics and Probability, vol. 1, 1961, pp. 547-561. [37] Román-Flores, H.; Flores-Franulič, A.; Chalco-Cano, Y., The fuzzy integral for monotone functions, Applied mathematics and computation, 185, 492-498, (2007) · Zbl 1116.26024 [38] Román-Flores, H.; Flores-Franulič, A.; Chalco-Cano, Y., A Jensen type inequality for fuzzy integrals, Information sciences, 177, 3192-3201, (2007) · Zbl 1127.28013 [39] Singh, R.P.; Kumar, Rajeev; Tuteja, R.K., Application of Hölder’s inequality in information theory, Information sciences, 152, 145-154, (2003) · Zbl 1042.94007 [40] Tian, J.-F., Inequalities and mathematical properties of uncertain variables, Fuzzy optimization and decision making, 10, 357-368, (2011) · Zbl 1254.28020 [41] Wong, F.-H.; Yeh, C.-C.; Yu, S.-L.; Hong, C.-H., Young’s inequality and related results on time scales, Applied mathematics letters, 18, 983-988, (2005) · Zbl 1080.26025 [42] Wu, S., Generalization of a sharp Hölder’s inequality and its application, Journal of mathematical analysis and applications, 332, 741-750, (2007) · Zbl 1120.26024 [43] Wu, S., A new sharpened and generalized version of Hölder’s inequality and its applications, Applied mathematics and computation, 197, 2, 708-714, (2008) · Zbl 1142.26018 [44] Xie, H.-B.; Zheng, Y.-P.; Guo, J.-Y.; Chen, X., Cross-fuzzy entropy: a new method to test pattern synchrony of bivariate time series, Information sciences, 180, 1715-1724, (2010) [45] Yang, W., A functional generalization of diamond-α integral Hölder’s nequality on time scales, Applied mathematics letters, 23, 1208-1212, (2010) · Zbl 1196.26038 [46] Zarezadeh, S.; Asadi, M., Results on residual Rényi entropy of order statistics and record values, Information sciences, 180, 21, 4195-4206, (2010) · Zbl 1204.94054
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.