Tian, Jingfeng; Hu, Xi-Mei A new reversed version of a generalized sharp Hölder’s inequality and its applications. (English) Zbl 1270.90104 Abstr. Appl. Anal. 2013, Article ID 901824, 9 p. (2013). Summary: We present a new reversed version of a generalized sharp Hölder’s inequality which is due to Wu and then give a new refinement of Hölder’s inequality. Moreover, the obtained result is used to improve the well-known Popoviciu-Vasić inequality. Finally, we establish the time scales version of the Beckenbach-type inequality. Cited in 1 ReviewCited in 8 Documents MSC: 90C48 Programming in abstract spaces Keywords:Hölder’s inequality; Popoviciu-Vasić inequality; Beckenbach-type inequality PDFBibTeX XMLCite \textit{J. Tian} and \textit{X.-M. Hu}, Abstr. Appl. Anal. 2013, Article ID 901824, 9 p. (2013; Zbl 1270.90104) Full Text: DOI References: [1] Abramovich, S.; Pečarić, J.; Varošanec, S., Sharpening Hölder’s and Popoviciu’s inequalities via functionals, The Rocky Mountain Journal of Mathematics, 34, 3, 793-810 (2004) · Zbl 1119.26022 [2] Ivanković, B.; Pečarić, J.; Varošanec, S., Properties of mappings related to the Minkowski inequality, Mediterranean Journal of Mathematics, 8, 4, 543-551 (2011) · Zbl 1232.26033 [3] Liu, B., Inequalities and convergence concepts of fuzzy and rough variables, Fuzzy Optimization and Decision Making, 2, 2, 87-100 (2003) · Zbl 1436.03279 [4] Nikolova, L.; Varošanec, S., Refinements of Hölder’s inequality derived from functions \(\psi_{p, q, \lambda}\) and \(\varphi_{p, q, \lambda}\), Annals of Functional Analysis, 2, 1, 72-83 (2011) · Zbl 1252.46016 [5] Buckley, S. M.; Koskela, P., Ends of metric measure spaces and Sobolev inequalities, Mathematische Zeitschrift, 252, 2, 275-285 (2006) · Zbl 1086.58021 [6] Franjić, I.; Khalid, S.; Pečarić, J., Refinements of the lower bounds of the Jensen functional, Abstract and Applied Analysis, 2011 (2011) · Zbl 1225.26041 [7] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge University Press · Zbl 0047.05302 [8] Hencl, S.; Koskela, P.; Zhong, X., Mappings of finite distortion: reverse inequalities for the Jacobian, The Journal of Geometric Analysis, 17, 2, 253-273 (2007) · Zbl 1126.30014 [9] Hu, K., On an inequality and its applications, Scientia Sinica, 24, 8, 1047-1055 (1981) · Zbl 0471.26009 [10] Jiang, R.; Koskela, P., Isoperimetric inequality from the Poisson equation via curvature, Communications on Pure and Applied Mathematics, 65, 8, 1145-1168 (2012) · Zbl 1247.53045 [11] Kuang, J., Applied Inequalities (2010), Jinan, China: Shandong Science and Technology Press, Jinan, China [12] Mićić, J.; Pavić, Z.; Pečarić, J., Extension of Jensen’s inequality for operators without operator convexity, Abstract and Applied Analysis, 2011 (2011) · Zbl 1221.47032 [13] Tian, J., Reversed version of a generalized sharp Hölder’s inequality and its applications, Information Sciences, 201, 61-69 (2012) · Zbl 1266.94012 [14] Tian, J., Inequalities and mathematical properties of uncertain variables, Fuzzy Optimization and Decision Making, 10, 4, 357-368 (2011) · Zbl 1254.28020 [15] Tian, J., Extension of Hu Ke’s inequality and its applications, Journal of Inequalities and Applications, 2011, article 77 (2011) · Zbl 1275.26044 [16] Tian, J., Property of a Hölder-type inequality and its application · Zbl 1277.26047 [17] Varošanec, S., A generalized Beckenbach-Dresher inequality and related results, Banach Journal of Mathematical Analysis, 4, 1, 13-20 (2010) · Zbl 1202.26040 [18] Wu, S. H., Generalization of a sharp Hölder’s inequality and its application, Journal of Mathematical Analysis and Applications, 332, 1, 741-750 (2007) · Zbl 1120.26024 [19] Beckenbach, E. F.; Bellman, R., Inequalities (1983), Berlin, Germany: Springer, Berlin, Germany [20] Aczél, J., Some general methods in the theory of functional equations in one variable. New applications of functional equations, Uspekhi Matematicheskikh Nauk, 11, 3, 3-68 (1956) [21] Popoviciu, T., On an inequality, Gazeta Matematica şi Fizica A, 11, 451-461 (1959) [22] Tian, J., Reversed version of a generalized Aczél’s inequality and its application, Journal of Inequalities and Applications, 2012, article 202 (2012) · Zbl 1279.26052 [23] Tian, J.; Wang, S., Refinements of generalized Aczel’s inequality and Bellman’s inequality and their applications, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.49020 [24] Vasić, P. M.; Pečarić, J. E., On the Hölder and some related inequalities, Mathematica, 25, 1, 95-103 (1983) · Zbl 0541.26008 [25] Wang, C.-L., Characteristics of nonlinear positive functionals and their applications, Journal of Mathematical Analysis and Applications, 95, 2, 564-574 (1983) · Zbl 0524.26012 [26] Anwar, M.; Bibi, R.; Bohner, M.; Pečarić, J., Integral inequalities on time scales via the theory of isotonic linear functionals, Abstract and Applied Analysis, 2011 (2011) · Zbl 1221.26026 [27] He, X.; Zhang, Q.-M., Lyapunov-type inequalities for some quasilinear dynamic system involving the \((p_1, p_2, \sum, p_m)\)-Laplacian on time scales, Journal of Applied Mathematics, 2010 (2011) · Zbl 1235.93187 [28] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 1-2, 18-56 (1990) · Zbl 0722.39001 [29] Saker, S. H., Some new inequalities of Opial’s type on time scales, Abstract and Applied Analysis, 2012 (2012) · Zbl 1242.26034 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.