Abstract
In this paper, we present some new inequalities for unitarily invariant norms involving Heron and Heinz means for matrices, which generalize the result of Theorem 2.1 (Fu and He in J. Math. Inequal. 7(4):727-737, 2013) and refine the inequality of Theorem 6 (Zhan in SIAM J. Matrix Anal. Appl. 20: 466-470, 1998). Our results are a refinement and a generalization of some existing inequalities. MSC:47A30, 15A60.
Highlights
Throughout, let Mm,n be the space of m × n complex matrices and Mn = Mn,n.A norm · is called unitarily invariant norm if UAV = A for all A ∈ Mn and for all unitary matrices U, V ∈ Mn
The first is the class of the Ky Fan k-norm · (k), defined as k
Where si(A) (i =, . . . , n) are the singular values of A with s (A) ≥ · · · ≥ sn(A), which are the eigenvalues of the positive semidefinite matrix
Summary
It is easy to see that as a function of v, Hv(a, b) is convex, attains its minimum at v The operator version of the Heinz mean [ ] asserts that if A, B and X are operators on a complex separable Hilbert space such that A and B are positive, for every unitarily invariant norm · , the function g(v) = AvXB –v + A –vXBv is convex on [ , ], attains its minimum at v Let A, B, X ∈ Mn, A, B are positive definite, Kaur and Singh [ ] have proved the following inequalities for any unitarily invariant norm · :
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