Mathematical analysis/Functional analysis
Reverses of operator Aczél inequality
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 475-481.

In this paper, we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of the operator Aczél inequality dealing with the weighted geometric mean.

Nous présentons dans cette Note des inégalités faisant intervenir des fonctions décroissantes sur les opérateurs et des moyennes d'opérateurs. Ces inégalités fournissent des inverses aux inégalités d'Aczél pour les opérateurs dans le cas des moyennes géométriques pondérées.

Published online:
DOI: 10.1016/j.crma.2018.04.005
Kaleibary, Venus 1; Furuichi, Shigeru 2

1 Department of Engineering, Basic Sciences Group, University of Science and Culture, Tehran, Iran
2 Department of Computer Science and System Analysis, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan
     author = {Kaleibary, Venus and Furuichi, Shigeru},
     title = {Reverses of operator {Acz\'el} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {475--481},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.005},
     language = {en},
     url = {}
AU  - Kaleibary, Venus
AU  - Furuichi, Shigeru
TI  - Reverses of operator Aczél inequality
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 475
EP  - 481
VL  - 356
IS  - 5
PB  - Elsevier
UR  -
DO  - 10.1016/j.crma.2018.04.005
LA  - en
ID  - CRMATH_2018__356_5_475_0
ER  - 
%0 Journal Article
%A Kaleibary, Venus
%A Furuichi, Shigeru
%T Reverses of operator Aczél inequality
%J Comptes Rendus. Mathématique
%D 2018
%P 475-481
%V 356
%N 5
%I Elsevier
%R 10.1016/j.crma.2018.04.005
%G en
%F CRMATH_2018__356_5_475_0
Kaleibary, Venus; Furuichi, Shigeru. Reverses of operator Aczél inequality. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 475-481. doi : 10.1016/j.crma.2018.04.005.

[1] Aczél, J. Some general methods in the theory of functional equations in one variable. New applications of functional equations, Usp. Mat. Nauk (N.S.), Volume 11 (1956), pp. 3-68 (in Russian)

[2] Ando, T.; Hiai, F. Operator log-convex functions and operator means, Math. Ann., Volume 350 (2011), pp. 611-630

[3] Bhatia, R. Matrix Analysis, Grad. Texts Math., vol. 169, Springer-Verlag, 1997

[4] Bourin, J.-C.; Lee, E.-Y.; Fujii, M.; Seo, Y. A matrix reverse Hölder inequality, Linear Algebra Appl., Volume 431 (2009), pp. 2154-2159

[5] Cho, Y.J.; Matić, M.; Pečarić, J. Improvements of some inequalities of Aczél's type, J. Math. Anal. Appl., Volume 259 (2001), pp. 226-240

[6] Dragomir, S.S. A generalization of Aczél's inequality in inner product spaces, Acta Math. Hung., Volume 65 (1994), pp. 141-148

[7] Dragomir, S.S. On new refinements and reverses of Young's operator inequality, RGMIA Res. Rep. Collect., Volume 18 (2015) (Art. 135)

[8] Furuichi, S. Further improvements of Young inequality, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (2017) | DOI

[9] Furuichi, S.; Moradi, H.R. Some refinements of classical inequalities, Rocky Mt. J. Math. (2018) (in press) | arXiv

[10] Furuta, T.; Mićić, J.; Pečarić, J.E.; Seo, Y. Mond–Pečarić Method in Operator Inequalities, Monogr. Inequal., vol. 1, Element, Zagreb, 2005

[11] Ghaemi, M.B.; Kaleibary, V. Some inequalities involving operator monotone functions and operator means, Math. Inequal. Appl., Volume 19 (2016), pp. 757-764

[12] Liao, W.; Wu, J.; Zhao, J. New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwan. J. Math., Volume 19 (2015), pp. 467-479

[13] Moslehian, M.S. Operator Aczél inequality, Linear Algebra Appl., Volume 434 (2011), pp. 1981-1987

[14] Popoviciu, T. On an inequality, Gaz. Mat. Fiz., Ser. A, Volume 11 (1959), pp. 451-461 (in Romanian)

[15] Specht, W. Zur Theorie der elementaren Mittel, Math. Z., Volume 74 (1960), pp. 91-98

[16] Tian, J.; Wang, W.-L. Generalizations of refined Hölder's inequalities and their applications, Math. Probl. Eng., Volume 2014 (2014)

[17] Tominaga, M. Specht's ratio in the Young inequality, Sci. Math. Jpn., Volume 55 (2002), pp. 583-588

Cited by Sources: