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Multiplicative inequalities for weighted geometric mean in Hermitian unital Banach \(*\)-algebras

Abstract

Consider the quadratic weighted geometric mean

$$\begin{aligned} x\circledS _{\nu }y:= \vert \vert yx^{-1} \vert ^{\nu }x \vert ^{2} \end{aligned}$$

for invertible elements xy in a Hermitian unital Banach \(*\) -algebra and real number \(\nu \). In this paper, by utilizing some results of Tominaga, Furuichi, Liao–Wu–Zhao, Zuo–Shi–Fujii and the author, we obtain various upper and lower bounds for the positive element \(\left( 1-\nu \right) \left| x\right| ^{2}+\nu \left| y\right| ^{2}\) in terms of \(x\circledS _{\nu }y,\) where \(\nu \in \left[ 0,1\right] ,\) under various assumptions for the elements xy involved. Applications for the classical weighted geometric mean

$$\begin{aligned} a\sharp _{\nu }b:=a^{1/2} ( a^{-1/2}ba^{-1/2}) ^{\nu }a^{1/2} \end{aligned}$$

of positive elements ab that satisfy the condition \(0<ka\le b\le Ka\) for certain numbers \(0<k<K,\) are also given.

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References

  1. Alzer, H., da Fonseca, C.M., Kovačec, A.: Young-type inequalities and their matrix analogues. Linear Multilinear Algebra 63(3), 622–635 (2015)

    MathSciNet  Article  Google Scholar 

  2. Bonsall, F.F., Duncan, J.: Complete Normed Algebra. Springer, New York (1973)

    Book  Google Scholar 

  3. Bullen, P.S.: Handbook of Mean and Their Inequalities. Kluwer Academic Publishers, Dordrecht (2003)

    Book  Google Scholar 

  4. Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990)

    MATH  Google Scholar 

  5. Dragomir, S.S.: Bounds for the normalized Jensen functional. Bull. Aust. Math. Soc. 74(3), 417–478 (2006)

    MathSciNet  Article  Google Scholar 

  6. Dragomir, S.S.: A note on Young’s inequality. RACSAM (2015). doi:10.1007/s13398-016-0300-8. http://rgmia.org/papers/v18/v18a126.pdf (to appear, preprint, RGMIA Res. Rep. Coll. 18, art. 126)

    MathSciNet  Article  Google Scholar 

  7. Dragomir, S.S.: A Note on new refinements and reverses of Young’s inequality. Transylv. J. Math. Mech. 8(1), 45–49 (2016). http://rgmia.org/papers/v18/v18a131.pdf [preprint, RGMIA Res. Rep. Coll. 18 (2015), Art. 131]

  8. Dragomir, S.S.: Quadratic weighted geometric mean in Hermitian unital Banach \(\ast \)-algebras. RGMIA Res. Rep. Coll. 19, 161 (2016). http://rgmia.org/papers/v19/v19a161.pdf [preprint]

  9. Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs. Victoria University (2000). http://rgmia.vu.edu.au/monographs

  10. Feng, B.Q.: The geometric means in Banach \(\ast \)-algebra. J. Oper. Theory 57(2), 243–250 (2007)

    MathSciNet  Google Scholar 

  11. Furuta, T.: Extension of the Furuta inequality and Ando–Hiai log-majorization. Linear Algebra Appl. 219, 139–155 (1995)

    MathSciNet  Article  Google Scholar 

  12. Furuichi, S.: Refined Young inequalities with Specht’s ratio. J. Egypt. Math. Soc. 20, 46–49 (2012)

    MathSciNet  Article  Google Scholar 

  13. Liao, W., Wu, J., Zhao, J.: New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant. Taiwan. J. Math. 19(2), 467–479 (2015)

    MathSciNet  Article  Google Scholar 

  14. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  15. Murphy, G.J.: \(C^{\ast }\)-Algebras and Operator Theory. Academic Press, New York (1990)

    MATH  Google Scholar 

  16. Okayasu, T.: The Löwner–Heinz inequality in Banach \(\ast \)-algebra. Glasgow Math. J. 42, 243–246 (2000)

    MathSciNet  Article  Google Scholar 

  17. Pečarić, J., Furuta, T., Mićić Hot, J., Seo, Y. : Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. Monographs in Inequalities, 1. Element, Zagreb (2005). ISBN: 953-197-572-8 (xiv + 262 pp. + loose errata)

  18. Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, New York (1992)

    MATH  Google Scholar 

  19. Shirali, S., Ford, J.W.M.: Symmetry in complex involutory Banach algebras, II. Duke Math. J. 37, 275–280 (1970)

    MathSciNet  Article  Google Scholar 

  20. Specht, W.: Zer Theorie der elementaren Mittel. Math. Z. 74, 91–98 (1960)

    MathSciNet  Article  Google Scholar 

  21. Tanahashi, K., Uchiyama, A.: The Furuta inequality in Banach \( \ast \)-algebras. Proc. Am. Math. Soc. 128, 1691–1695 (2000)

    MathSciNet  Article  Google Scholar 

  22. Tominaga, M.: Specht’s ratio in the Young inequality. Sci. Math. Jpn. 55, 583–588 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Zuo, G., Shi, G., Fujii, M.: Refined Young inequality with Kantorovich constant. J. Math. Inequal. 5, 551–556 (2011)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referees for their valuable comments that have been implemented in the final version of the paper.

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Correspondence to S. S. Dragomir.

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Dragomir, S.S. Multiplicative inequalities for weighted geometric mean in Hermitian unital Banach \(*\)-algebras. RACSAM 112, 1349–1365 (2018). https://doi.org/10.1007/s13398-017-0430-7

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  • DOI: https://doi.org/10.1007/s13398-017-0430-7

Keywords

  • Weighted geometric mean
  • Young’s inequality
  • Operator modulus
  • Arithmetic mean-geometric mean inequality
  • Hermitian unital Banach \(*\)-algebra

Mathematics Subject Classification

  • 47A63
  • 47A30
  • 15A60
  • 26D15
  • 26D10