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Refinements of the heinz inequalities for operators and matrices

  • Mahdi Mohammadi Gohari und Maryam Amyari EMAIL logo
Veröffentlicht/Copyright: 20. November 2018
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 68 Heft 6

Abstract

Suppose that A, B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that

A#B≤11−2μA12Fμ(A−12BA−12)A12 ≤12[A#B+Hμ(A,B)] ≤12[11−2μA12Fμ(A−12BA−12)A12+Hμ(A,B)] ≤⋯≤12nA#B+2n−12nHμ(A,B) ≤12n(1−2μ)A12Fμ(A−12BA−12)A12+2n−12nHμ(A,B) ≤12n+1A#B+2n+1−12n+1Hμ(A,B) ≤⋯≤Hμ(A,B)

for each μ∈[0,1]∖{12}, where Hμ (A, B) and A#B are the Heinz mean and the geometric mean for operators A, B, respectively, and Fμ∈C(sp(A−12BA−12)) is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.

MSC 2010: Primary 47A63; 15A60; Secondary 26A51
Keywords: convex function; Heinz operator inequalities; Heinz mean; Hermite-Hadamard inequality; refinement; unitarily invariant norm

  1. (Communicated by Werner Timmermann)

References

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Received: 2017-06-20
Accepted: 2017-09-14
Published Online: 2018-11-20
Published in Print: 2018-12-19

© 2018 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Idempotents, group membership and their applications
  2. Residuation in non-associative MV-algebras
  3. An extension of F. Šik’s theorem on modular lattices
  4. Weak pseudo-BCK algebras
  5. Witt functor of a quadratic order
  6. A modification of a problem of Diophantus
  7. Cauchy problems involving a Hadamard-type fractional derivative
  8. Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
  9. Sturm-Picone comparison theorems for nonlinear impulsive differential equations
  10. On oscillatory fourth order nonlinear neutral differential equations – III
  11. Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
  12. On the representation of involutive Jamesian functions
  13. Refinements of the heinz inequalities for operators and matrices
  14. Generalizations of Reid inequality
  15. Probabilistic convergence transformation groups
  16. Cluster sets and topology
  17. Some characterizations for Markov processes as mixed renewal processes
  18. Equivalent conditions of complete convergence for weighted sums of ANA random variables
https://doi.org/10.1515/ms-2017-0192
Schlagwörter für diesen Artikel
convex function; Heinz operator inequalities; Heinz mean; Hermite-Hadamard inequality; refinement; unitarily invariant norm

Artikel in diesem Heft

  1. Idempotents, group membership and their applications
  2. Residuation in non-associative MV-algebras
  3. An extension of F. Šik’s theorem on modular lattices
  4. Weak pseudo-BCK algebras
  5. Witt functor of a quadratic order
  6. A modification of a problem of Diophantus
  7. Cauchy problems involving a Hadamard-type fractional derivative
  8. Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
  9. Sturm-Picone comparison theorems for nonlinear impulsive differential equations
  10. On oscillatory fourth order nonlinear neutral differential equations – III
  11. Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
  12. On the representation of involutive Jamesian functions
  13. Refinements of the heinz inequalities for operators and matrices
  14. Generalizations of Reid inequality
  15. Probabilistic convergence transformation groups
  16. Cluster sets and topology
  17. Some characterizations for Markov processes as mixed renewal processes
  18. Equivalent conditions of complete convergence for weighted sums of ANA random variables
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