# More accurate Young, Heinz, and Hölder inequalities for matrices

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## Abstract

In this paper we deal with a more precise estimates for the matrix versions of Young, Heinz, and Hölder inequalities. First we give an improvement of the matrix Heinz inequality for the case of the Hilbert–Schmidt norm. Then, we refine matrix Young-type inequalities for the case of Hilbert–Schmidt norm, which hold under certain assumptions on positive semidefinite matrices appearing therein. Finally, we give the refinement and the reverse of the matrix Hölder inequality which holds for every unitarily invariant norm. As applications, we also obtain improvements of some well-known matrix inequalities in a quotient form. Our results are compared with the previously known from the literature.

## Keywords

Young inequality Heinz inequality Hölder inequality Hilbert–Schmidt norm Unitarily invariant norm Positive semidefinite matrix## Mathematics Subject Classification

15A42 15A60 26D20## References

- 1.T. Ando, Matrix Young inequality. Oper. Theory Adv. Appl.
**75**, 33–38 (1995)Google Scholar - 2.R. Bhatia, C. Davis, More matrix forms of the arithmetic–geometric mean inequality. SIAM J. Matrix Anal. Appl.
**14**, 132–136 (1993)MathSciNetCrossRefMATHGoogle Scholar - 3.R. Bhatia,
*Matrix Analysis*(Springer, New York, 1997)CrossRefGoogle Scholar - 4.R. Bhatia, K.R. Parthasarathy, Positive definite functions and operator inequalities. Bull. Lond. Math. Soc.
**32**, 214–228 (2000)MathSciNetCrossRefMATHGoogle Scholar - 5.R. Bhatia, Interpolating the arithmetic–geometric mean inequality and its operator version. Linear Algebra Appl.
**413**, 355–363 (2006)MathSciNetCrossRefMATHGoogle Scholar - 6.O. Hirzallah, F. Kittaneh, Matrix Young inequalities for the Hilbert–Schmidt norm. Linear Algebra Appl.
**308**, 77–84 (2000)MathSciNetCrossRefMATHGoogle Scholar - 7.R.A. Horn, C.R. Johnson,
*Matrix Analysis*(Cambridge University Press, New York, 1985)CrossRefMATHGoogle Scholar - 8.F. Kittaneh, Norm inequalities for fractional powers of positive operators. Lett. Math. Phys.
**27**, 279–285 (1993)MathSciNetCrossRefMATHGoogle Scholar - 9.F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrices. J. Math. Anal. Appl.
**361**, 262–269 (2010)MathSciNetCrossRefMATHGoogle Scholar - 10.F. Kittaneh, Y. Manasrah, Reverse Young and Heinz inequalities for matrices. Linear Multilinear Algebra
**59**, 1031–1037 (2011)MathSciNetCrossRefMATHGoogle Scholar - 11.F. Kittaneh, M. Krnić, Refined Heinz operator inequalities. Linear Multilinear Algebra
**61**, 1148–1157 (2013)MathSciNetCrossRefMATHGoogle Scholar - 12.H. Kosaki, Arithmetic–geometric mean and related inequalities for operators. J. Funct. Anal.
**156**, 429–451 (1998)MathSciNetCrossRefMATHGoogle Scholar - 13.M. Krnić, N. Lovričević, J. Pečarić, Jessen’s functional, its properties and applications. An. Şt. Univ. Ovidius Constanţa
**20**, 225–248 (2012)MATHGoogle Scholar - 14.M. Krnić, On some refinements and converses of multidimensional Hilbert-type inequalities. Bull. Aust. Math. Soc.
**85**, 380–394 (2012)MathSciNetCrossRefMATHGoogle Scholar - 15.M. Krnić, N. Lovričević, J. Pečarić, On the properties of McShane’s functional and their applications. Period. Math. Hung.
**66**(2), 159–180 (2013)CrossRefMATHGoogle Scholar - 16.B. Simon,
*Trace Ideals and Their Applications*, London Mathematical Society Lecture Note Series, 35 (Cambridge University Press, Cambridge, 1979)MATHGoogle Scholar

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© Akadémiai Kiadó, Budapest, Hungary 2015