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Some inequalities involving Heron and Heinz means of two convex functionals. (English) Zbl 1463.47054

For \(\lambda \in[0,1]\), let \(a\nabla_\lambda b=(1-\lambda)a+\lambda b\) and \(a\sharp_\lambda b=a^{1-\lambda}b^\lambda \) be the \(\lambda\)-weighted arithmetic and geometric means of the numbers \(a,b>0\), respectively. The Heron and Heinz means of these numbers are defined by \(K_\lambda (a,b)=(1-\lambda)\sqrt{ab}+\lambda \frac{a+b}2\) and \(HZ_\lambda (a,b)=2^{-1}\big(a^{1-\lambda} b^\lambda+ a^{\lambda}b^{1-\lambda}\big)\), respectively.
Many inequalities are known to hold between these means as well as between their operator analogs. The authors consider in this paper some functional analytic versions of these means by replacing \(a,b\) by two convex lsc proper functions \(f,g:H\to\mathbb{R}\cup\{\infty\}\), where \(H\) is a Hilbert space, as they were introduced in M. Raïssouli and H. Bouziane [Ann. Sci. Math. Qué. 30, No. 1, 79–107 (2006; Zbl 1222.49025)]. The functional arithmetic and geometric means are defined by \(\mathcal{A}_\lambda(f,g)=(1-\lambda)f+\lambda g\) and \[ \mathcal{G}_\lambda(f,g)=\frac{\sin(\pi\lambda)}{\pi}\int_0^1t^{\lambda-1}(1-t)^{-\lambda}\big((1-t)f^*+tg^*\big)^*dt, \] respectively, where \(h^*\) denotes the Fenchel conjugate of a convex lsc proper function \(h:H\to\mathbb{R}\cup\{\infty\},\, h^*(x^*)=\sup\{\operatorname{Re} e\langle x,x^*\rangle -h(x): x\in H\},\, x^*\in H\).
The following inequalities hold: \(\mathcal{H}_\lambda(f,g)\le \mathcal{G}_\lambda(f,g)\le \mathcal{A}_\lambda(f,g),\) where \(\mathcal{H}_\lambda(f,g)=\big((1-t)f^*+tg^*\big)^*\) is the \(\lambda\)-weighted functional harmonic mean of \(f,g\).
The functional Heron and Heinz means are defined by \(\mathcal{K}_\lambda(f,g)=(1-\lambda)\mathcal{G}(f,g)+\lambda \mathcal{A}(f,g)\) and \(\mathcal{HZ}_\lambda=2^{-1}\big(\mathcal{G}_\lambda(f,g)+\mathcal{G}_{1-\lambda}(f,g)\big)\), respectively.
The authors prove some inequalities involving these means as well as for their operator versions. As they mention in the abstract: “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.”

MSC:

47A64 Operator means involving linear operators, shorted linear operators, etc.
47A63 Linear operator inequalities
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics

Citations:

Zbl 1222.49025
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References:

[1] Bhatia, R., Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl., 413, 355-363 (2006) · Zbl 1092.15018 · doi:10.1016/j.laa.2005.03.005
[2] Dinh, T. H.; Dumitru, R.; Franco, J. A., The matrix power means and interpolations, Adv. Oper. Theory, 3, 647-654 (2018) · Zbl 06902458 · doi:10.15352/aot.1801-1288
[3] Dragomir, S. S.; Pearce, C. E M., Selected Topics on Hermite-Hadamard Inequalities and Applications (2000)
[4] Farissi, A., Simple proof and refinement of Hermite-Hadamard inequality, J. Math. Inequal., 4, 365-369 (2010) · Zbl 1197.26017 · doi:10.7153/jmi-04-33
[5] Fujii, M.; Furuichi, S.; Nakamoto, R., Estimations of Heron means for positive operators, J. Math. Inequal., 10, 19-30 (2016) · Zbl 1385.47009 · doi:10.7153/jmi-10-02
[6] Furuichi, S., Refined Young inequalities with Specht’s ratio, J. Egypt. Math. Soc., 20, 46-49 (2012) · Zbl 1287.47031 · doi:10.1016/j.joems.2011.12.010
[7] Ito, M., Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator means, Adv. Oper. Theory (2018) · Zbl 06946376
[8] Khosravi, M., Some matrix inequalities for weighted power mean, Ann. Funct. Anal., 7, 348-357 (2016) · Zbl 1337.15020 · doi:10.1215/20088752-3544480
[9] 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. · Zbl 06804309
[10] Kittaneh, F.; Manasrah, Y., Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 36, 262-269 (2010) · Zbl 1180.15021 · doi:10.1016/j.jmaa.2009.08.059
[11] Kittaneh, F.; Manasrah, Y., Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra, 59, 1031-1037 (2011) · Zbl 1225.15022 · doi:10.1080/03081087.2010.551661
[12] Kittaneh, F.; Moslehian, M. S.; Sababheh, M., Quadratic interpolation of the Heinz mean, Math. Inequal. Appl., 21, 739-757 (2018) · Zbl 1402.15016
[13] Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 205-224 (1980) · Zbl 0412.47013 · doi:10.1007/BF01371042
[14] Liang, J.; Shi, G., Refinements of the Heinz operator inequalities, Linear Multilinear Algebra, 63, 1337-1344 (2015) · Zbl 1321.47039 · doi:10.1080/03081087.2014.936434
[15] Minculete, N., A result about Young’s inequality and several applications, Sci. Magna, 7, 61-68 (2011)
[16] Raïssouli, M.; Bouziane, H., Arithmetico-geometrico-harmonic functional mean in convex analysis, Ann. Sci. Math. Qu´ebec, 30, 79-107 (2006) · Zbl 1222.49025
[17] Raïssouli, M., Functional versions of some refined and reversed operator meaninequalities, Bull. Aust. Math. Soc., 96, 496-503 (2017) · Zbl 1485.47021 · doi:10.1017/S0004972717000594
[18] Raïssouli, M.; Furuichi, S., Functional version for Furuta parametric relative operator entropy, J. Inequal. Appl. (2018) · Zbl 1498.47041
[19] Tominaga, M., Specht’s ratio in the Young inequality, Sci. Math. Jpn., 55, 538-588 (2002) · Zbl 1021.47010
[20] Yang, C.; Ren, Y., Some results of Heron mean and Young’s inequalities, J. Inequal. Appl. (2018) · Zbl 1498.47045
[21] Zou, L., Inequalities related to Heinz and Heron means, J. Math. Inequal., 7, 389-397 (2013) · Zbl 1274.15023
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