Abstract
In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach in a fast and nice way.
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References
R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl., 413 (2006), 355–363.
T. H. Dinh, R. Dumitru and J. A. Franco, The matrix power means and interpolations, Adv. Oper. Theory, 3 (2018), 647–654.
S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs (2000) (rgmia.org/monographs/hermitehadamard.html).
A. Farissi, Simple proof and refinement of Hermite–Hadamard inequality, J. Math. Inequal., 4 (2010), 365–369.
M. Fujii, S. Furuichi, and R. Nakamoto, Estimations of Heron means for positive operators, J. Math. Inequal., 10 (2016), 19–30.
S. Furuichi, Refined Young inequalities with Specht’s ratio, J. Egypt. Math. Soc., 20 (2012), 46–49.
M. Ito, Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator means, Adv. Oper. Theory (2018), doi.org/10.15352/aot.1801.1303.
M. Khosravi, Some matrix inequalities for weighted power mean, Ann. Funct. Anal., 7 (2016), 348–357.
Y. Kapil, C. Conde, M. S. Moslehian, M. Singh and M. Sababheh, Norm inequalities related to the Heron and Heinz means, Mediterr. J. Math., 14 (2017), Art. 213, 18 pp.
F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 36 (2010), 262–269.
F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra, 59 (2011), 1031–1037.
F. Kittaneh, M. S. Moslehian and M. Sababheh, Quadratic interpolation of the Heinz mean, Math. Inequal. Appl., 21 (2018), 739–757.
F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), 205–224.
J. Liang and G. Shi, Refinements of the Heinz operator inequalities, Linear Multilinear Algebra, 63 (2015), 1337–1344.
N. Minculete, A result about Young’s inequality and several applications, Sci. Magna, 7 (2011), 61–68.
M. Raïssouli and H. Bouziane, Arithmetico-geometrico-harmonic functional mean in convex analysis, Ann. Sci. Math. Qu´ebec, 30 (2006), 79–107.
M. Raïssouli, Functional versions of some refined and reversed operator meaninequalities, Bull. Aust. Math. Soc., 96, (2017), 496–503.
M. Raïssouli and S. Furuichi, Functional version for Furuta parametric relative operator entropy, J. Inequal. Appl., 2018, Paper No. 212, 10 pp.
M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Jpn., 55 (2002), 538–588.
C. Yang and Y. Ren, Some results of Heron mean and Young’s inequalities, J. Inequal. Appl., 2018, Paper No. 172, 9 pp.
L. Zou, Inequalities related to Heinz and Heron means, J. Math. Inequal., 7 (2013), 389–397.
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The authors thank anonymous referees for valuable comments and suggestions to improve our manuscript.
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The second named author was partially supported by JSPS KAKENHI Grant #16K05257.
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Raïssouli, M., Furuichi, S. Some inequalities involving Heron and Heinz means of two convex functionals. Anal Math 46, 345–365 (2020). https://doi.org/10.1007/s10476-020-0026-x
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DOI: https://doi.org/10.1007/s10476-020-0026-x