Refined converses of Jensen’s inequality for operators
Tóm tắt
In this paper converses of a generalized Jensen’s inequality for a continuous field of self-adjoint operators, a unital field of positive linear mappings and real-valued continuous convex functions are studied. New refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions. MSC:47A63, 47A64.
Tài liệu tham khảo
Davis C: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc. 1957, 8: 42–44. 10.1090/S0002-9939-1957-0084120-4
Choi MD:A Schwarz inequality for positive linear maps on C ∗ -algebras. Ill. J. Math. 1974, 18: 565–574.
Hansen F, Pedersen GK: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 1982, 258: 229–241. 10.1007/BF01450679
Hansen F, Pedersen GK: Jensen’s operator inequality. Bull. Lond. Math. Soc. 2003, 35: 553–564. 10.1112/S0024609303002200
Mond B, Pečarić J: On Jensen’s inequality for operator convex functions. Houst. J. Math. 1995, 21: 739–754.
Furuta T, Mićić Hot J, Pečarić J, Seo Y Monographs in Inequalities 1. In Mond-Pečarić Method in Operator Inequalities. Element, Zagreb; 2005.
Hansen F, Pečarić J, Perić I: Jensen’s operator inequality and its converses. Math. Scand. 2007, 100: 61–73.
Abramovich S, Jameson G, Sinnamon G: Refining Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roum. 2004, 47: 3–14.
Dragomir SS: A new refinement of Jensen’s inequality in linear spaces with applications. Math. Comput. Model. 2010, 52: 1497–1505. 10.1016/j.mcm.2010.05.035
Fujii JI: An external version of the Jensen operator inequality. Sci. Math. Japon. Online 2011, 2011: 59–62.
Fujii JI, Pečarić J, Seo Y: The Jensen inequality in an external formula. J. Math. Inequal. 2012, 6: 473–480.
Ivelić A, Matković A, Pečarić JE: On a Jensen-Mercer operator inequality. Banach J. Math. Anal. 2011, 5: 19–28.
Khosravi M, Aujla JS, Dragomir SS, Moslehian MS: Refinements of Choi-Davis-Jensen’s inequality. Bull. Math. Anal. Appl. 2011, 3: 127–133.
Mićić J, Pavić Z, Pečarić J: Extension of Jensen’s operator inequality for operators without operator convexity. Abstr. Appl. Anal. 2011, 2011: 1–14.
Mićić J, Pečarić J, Perić J: Extension of the refined Jensen’s operator inequality with condition on spectra. Ann. Funct. Anal. 2012, 3: 67–85.
Moslehian MS, Kian M: Jensen type inequalities for Q -class functions. Bull. Aust. Math. Soc. 2011, 85: 128–142.
Rooin J: A refinement of Jensen’s inequality. J. Inequal. Pure Appl. Math. 2005., 6(2): Article ID 38
Srivastava HM, Xia ZG, Zhang ZH: Some further refinements and extensions of the Hermite-Hadamard and Jensen inequalities in several variables. Math. Comput. Model. 2011, 54: 2709–2717. 10.1016/j.mcm.2011.06.057
Xiao ZG, Srivastava HM, Zhang ZH: Further refinements of the Jensen inequalities based upon samples with repetitions. Math. Comput. Model. 2010, 51: 592–600. 10.1016/j.mcm.2009.11.004
Wang LC, Ma XF, Liu LH: A note on some new refinements of Jensen’s inequality for convex functions. J. Inequal. Pure Appl. Math. 2009., 10(2): Article ID 48
Mićić J, Pavić Z, Pečarić J: Jensen’s inequality for operators without operator convexity. Linear Algebra Appl. 2011, 434: 1228–1237. 10.1016/j.laa.2010.11.004
Mićić J, Pečarić J, Perić J: Refined Jensen’s operator inequality with condition on spectra. Oper. Matrices 2013, 7: 293–308.
Mond B, Pečarić JE: Converses of Jensen’s inequality for linear maps of operators. An. Univ. Timiş., Ser. Mat.-Inform. 1993, 2: 223–228.
Mond B, Pečarić J: Converses of Jensen’s inequality for several operators. Rev. Anal. Numér. Théor. Approx. 1994, 23: 179–183.
Furuta T: Operator inequalities associated with Hölder-McCarthy and Kantorovich inequalities. J. Inequal. Appl. 1998, 2: 137–148.
Mićić J, Seo Y, Takahasi SE, Tominaga M: Inequalities of Furuta and Mond-Pečarić. Math. Inequal. Appl. 1999, 2: 83–111.
Mićić J, Pečarić J, Seo Y, Tominaga M: Inequalities of positive linear maps on Hermitian matrices. Math. Inequal. Appl. 2000, 3: 559–591.
Mićić J, Pečarić J, Seo Y: Converses of Jensen’s operator inequality. Oper. Matrices 2010, 4: 385–403.
Mićić J, Pavić Z, Pečarić J: Some better bounds in converses of the Jensen operator inequality. Oper. Matrices 2012, 6: 589–605.
Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht; 1993.