×

Some inequalities involving interpolations between arithmetic and geometric mean. (English) Zbl 1518.26019

Summary: In this article, we mainly study the interpolations between arithmetic mean and geometric mean - power mean, Heron mean and Heinz mean. First, we obtain the improvement and reverse improvement of arithmetic-power mean inequalities by the convexity of the function. We show that the proof of Heron mean inequality due to C. Yang and Y. Ren [J. Inequal. Appl. 2018, Paper No. 172, 9 p. (2018; Zbl 1498.47045)], is not substantial. In addition, we also obtain Heron-Heinz mean inequalities for \(t \in \mathbb{R}\). Further corresponding operator versions and generalizations are also established.

MSC:

26E60 Means
47A64 Operator means involving linear operators, shorted linear operators, etc.

Citations:

Zbl 1498.47045
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ANDO, T.: Topics on operator inequalities, Lecture Note, Sapporo, (1978). · Zbl 0388.47024
[2] BHATIA, R.: Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl. 413 (2006), 355-363. · Zbl 1092.15018
[3] KUBO, F.—ANDO, T.: Means of positive linear operators, Math. Ann. 246 (1980), 205-224. · Zbl 0412.47013
[4] KITTANEH, F.—MANASRAH, Y.: Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361 (2010), 262-269. · Zbl 1180.15021
[5] KITTANEH, F.—MANASRAH, Y.: Reversed Young and Heinz inequalities for matrices, Linear Multilinear Algebra 59 (2011), no. 9, 1031-1037. · Zbl 1225.15022
[6] KITTANEH, F.—MOSLEHIAN, M. S.—SABABHEH, M.: Quadratic interpolation of the Heinz means, Math. Inequal. Appl. 21 (2018), no. 3, 739-757. · Zbl 1402.15016
[7] MARSHALL, A. W.—OLKIN, I.—ARNOLD, B. C.: : Inequalities: Theory of Majorization and its Application. Second edition. Springer Series in Statistics. Springer, New York, 2011. · Zbl 1219.26003
[8] PUSZ, W.—WORONOWICZ, S. L.: Functional calculus for sesquilinear forms and the purification map,Rep. Math. Phys. 8 (1975), 159-170. · Zbl 0327.46032
[9] SABABHEH, M.: Means refinements via convexity, Mediterr. J. Math. 14 (2017) no. 3, paper no. 25, 16 pp. · Zbl 06802130
[10] SABABHEH, M.—FURUICHI, S.—HEYDARBEYGI, Z.—MORADI, H. R.: On the arithemetic-geometric mean inequality, J. Math. Inequal. 15 (2021), no. 3, 1255-1266. · Zbl 1480.26024
[11] SABABHEH, M.—MORADI, H. R.: Radical convex functions, Mediterr. J. Math. 18 (2021), no. 4, paper no. 137, 15. pp. · Zbl 1473.26009
[12] YANG, C.—REN, Y.: : Some results of Heron mean and Young’s inequalities,J. Inequal. Appl. 2018 (2018), paper no, 172, 9 pp. · Zbl 1498.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.