Rocky Mountain Journal of Mathematics

Some refinements of classical inequalities

Shigeru Furuichi and Hamid Reza Moradi

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We give some new refinements and reverses of Young inequalities in both additive and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some results relevant to the Heron mean are also considered.

Article information

Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2289-2309.

First available in Project Euclid: 14 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 46L05: General theory of $C^*$-algebras 47A60: Functional calculus

Operator inequality Hermite-Hadamard inequality Young inequality Heron mean


Furuichi, Shigeru; Moradi, Hamid Reza. Some refinements of classical inequalities. Rocky Mountain J. Math. 48 (2018), no. 7, 2289--2309. doi:10.1216/RMJ-2018-48-7-2289.

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  • R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Lin. Alg. Appl. \bf413 (2006), 355–363.
  • ––––, Positive definite matrices, Princeton University Press, Princeton, 2007.
  • R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Lin. Alg. Appl. \bf308 (2000), 203–211.
  • S.S. Dragomir, A note on Young's inequality, Rev. Roy. Acad. Cienc. Exact. \bf111 (2017), 349–354.
  • A. El Farissi, Simple proof and refinement of Hermite-Hadamard inequality, J. Math. Inequal. \bf4 (2010), 365–369.
  • Y. Feng, Refinements of the Heinz inequalities, J. Inequal. Appl. (2012), Art. no. 18.
  • M. Fujii, S. Furuichi and R. Nakamoto, Estimations of Heron means for positive operators, J. Math. Inequal. \bf10 (2016), 19–30.
  • S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal. \bf5 (2011), 21–31.
  • ––––, Refined Young inequalities with Specht's ratio, J. Egyptian Math. Soc. \bf20 (2012), 46–49.
  • ––––, Operator inequalities among arithmetic mean, geometric mean and harmonic mean, J. Math. Inequal. \bf8 (2014), 669–672.
  • S. Furuichi and N. Minculete, Alternative reverse inequalities for Young's inequality, J. Math Inequal. \bf5 (2011), 595–600.
  • T. Furuta and M. Yanagida, Generalized means and convexity of inversion for positive operators, Amer. Math. Month. \bf105 (1998), 258–259.
  • F. Hiai, Matrix analysis: Matrix monotone functions, matrix means, and majorization, Interdisc. Info. Sci. \bf46 (2010), 139–248.
  • F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Lin. Multilin. Alg. \bf59 (2011), 1031–1037.
  • W. Liao, J. Wu and J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. \bf19 (2015), 467–479.
  • M. Lin, On an operator Kantorovich inequality for positive linear maps, J. Math. Anal. Appl. \bf402 (2013), 127–132.
  • M. Lin, Squaring a reverse AM-GM inequality, Stud. Math. \bf215 (2013), 189–194.
  • H.R. Moradi, S. Furuichi and N. Minculete, Estimates for Tsallis relative operator entropy, Math. Inequal. Appl. \bf20 (2017), 1079–1088.
  • C.P. Niculescu and L.E. Persson, Old and new on the Hermite-Hadamard inequality, Real Anal. Exch. \bf29 (2004), 663–686.
  • M. Sababheh and D. Choi, A complete refinement of Young's inequality, J. Math. Anal. Appl. \bf440 (2016), 379–393.
  • M. Sababheh and M.S. Moslehian, Advanced refinements of Young and Heinz inequalities, J. Num. Th. \bf172, 178–199.
  • L. Wang, On extensions and refinements of Hermite-Hadamard inequalities for convex functions, Math. Inequal. Appl. \bf6 (2003), 659–666.
  • K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Lin. Alg. Appl. \bf394 (2005), 109–118.
  • H. Zuo, G. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. \bf5 (2011), 551–556.