# Improved Heinz inequalities via the Jensen functional

Open Mathematics (2013)

- Volume: 11, Issue: 9, page 1698-1710
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topMario Krnić, and Josip Pečarić. "Improved Heinz inequalities via the Jensen functional." Open Mathematics 11.9 (2013): 1698-1710. <http://eudml.org/doc/269202>.

@article{MarioKrnić2013,

abstract = {By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.},

author = {Mario Krnić, Josip Pečarić},

journal = {Open Mathematics},

keywords = {Heinz mean; Heinz norm inequalities; Heinz operator inequalities; Jensen functional; Monotonicity; Convex function; Unitarily invariant norm; Refinement; monotonicity; convex function; unitarily invariant norm; refinement},

language = {eng},

number = {9},

pages = {1698-1710},

title = {Improved Heinz inequalities via the Jensen functional},

url = {http://eudml.org/doc/269202},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Mario Krnić

AU - Josip Pečarić

TI - Improved Heinz inequalities via the Jensen functional

JO - Open Mathematics

PY - 2013

VL - 11

IS - 9

SP - 1698

EP - 1710

AB - By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.

LA - eng

KW - Heinz mean; Heinz norm inequalities; Heinz operator inequalities; Jensen functional; Monotonicity; Convex function; Unitarily invariant norm; Refinement; monotonicity; convex function; unitarily invariant norm; refinement

UR - http://eudml.org/doc/269202

ER -

## References

top- [1] Bhatia R., Matrix Analysis, Grad. Texts in Math., 169, Springer, New York, 1997
- [2] Bhatia R., Positive Definite Matrices, Princeton Ser. Appl. Math., Princeton University Press, Princeton, 2007 Zbl1125.15300
- [3] Bhatia R., Davis C., More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl., 1993, 14(1), 132–136 http://dx.doi.org/10.1137/0614012 Zbl0767.15012
- [4] Dragomir S.S., Pečarić J., Persson L.E., Properties of some functionals related to Jensen’s inequality, Acta Math. Hungar., 1996, 70(1–2), 129–143 http://dx.doi.org/10.1007/BF00113918 Zbl0847.26013
- [5] Furuta T., Mićić Hot J., Pečarić J., Seo Y., Mond-Pečaric Method in Operator Inequalities, Monographs in Inequalities, 1, Element, Zagreb, 2005 Zbl1135.47012
- [6] Hiai F., Kosaki H., Means for matrices and comparison of their norms, Indiana Univ. Math. J., 1999, 48(3), 899–936 http://dx.doi.org/10.1512/iumj.1999.48.1665 Zbl0934.15023
- [7] Kittaneh F., On the convexity of the Heinz means, Integral Equations Operator Theory, 2010, 68(4), 519–527 http://dx.doi.org/10.1007/s00020-010-1807-6 Zbl1230.47026
- [8] Kittaneh F., Krnić M., Lovričević N., Pečarić J., Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. Debrecen, 2012, 80(3–4), 465–478 http://dx.doi.org/10.5486/PMD.2012.5193 Zbl1275.47038
- [9] Kittaneh F., Manasrah Y., Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 2010, 361(1), 262–269 http://dx.doi.org/10.1016/j.jmaa.2009.08.059 Zbl1180.15021
- [10] Kittaneh F., Manasrah Y., Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra, 2011, 59(9), 1031–1037 http://dx.doi.org/10.1080/03081087.2010.551661 Zbl1225.15022
- [11] Klaričić Bakula M., Matić M., Pečarić J., On inequalities complementary to Jensen’s inequality, Mat. Bilten, 2008, 32, 17–27 Zbl1265.26073
- [12] Krnić M., Lovričević N., Pečarić J., Jensen’s operator and applications to mean inequalities for operators in Hilbert space, Bull. Malays. Math. Sci. Soc., 2012, 35(1), 1–14 Zbl1248.47018
- [13] Kubo F., Ando T., Means of positive linear operators, Math. Ann., 1979/80, 246(3), 205–224 http://dx.doi.org/10.1007/BF01371042 Zbl0412.47013
- [14] Mitrinović D.S., Pečarić J.E., Fink A.M., Math. Appl. (East European Ser.), 61, Classical and New Inequalities in Analysis, Kluwer, Dordrecht, 1993 Zbl0771.26009
- [15] Simon B., Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser., 35, Cambridge University Press, Cambridge-New York, 1979 Zbl0423.47001