Wigner-Yanase-Dyson function and logarithmic mean. (English) Zbl 07879377
Summary: The ordering between Wigner-Yanase-Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner-Yanase-Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequalities based on the obtained results.
Keywords:
Wigner-Yanase-Dyson function; logarithmic mean; Kantorovich constant; Specht ratio and reverse inequalitiesReferences:
| [1] | From Theorem 2.3, we also have the following corollary. COROLLARY 4.2. Let S and T be positive operators with αS T β S for |
| [2] | < α β and let 0 p 1 . Then we have W p (S, T ) k p • L(S, T ), k p := max α x β K(x) p(1-p) . |
| [3] | S. FURUICHI, Unitarily invariant norm inequalities for some means, J. Inequal. Appl., 2014 (2014), Art. 158. · Zbl 1379.15012 |
| [4] | S. FURUICHI AND M. E. AMLASHI, On bounds of logarithmic mean and mean inequality chain, arXiv:2203.01134. |
| [5] | S. FURUICHI AND H. R. MORADI, Advances in mathematical inequalities, De Gruyter, 2020. · Zbl 1447.26017 |
| [6] | S. FURUICHI AND K. YANAGI, Schrödinger uncertainty relation, Wigner-Yanase-Dyson skew infor-mation and metric adjusted correlation measure, J. Math. Anal. Appl., 388 (2) (2012), 1147-1156. · Zbl 1326.81036 |
| [7] | P. GIBILISCO, F. HANSEN AND T. ISOLA, On a correspondence between regular and non-regular operator monotone functions, Linear Algebra Appl., 430 (2009), 2225-2232. · Zbl 1178.47011 |
| [8] | F. HANSEN, Metric adjusted skew information, Proc. Nat. Acad. Sci., 105 (2008), 9909-9916. · Zbl 1205.94058 |
| [9] | F. HIAI AND H. KOSAKI, Means for matrices and comparison of their norms, Indiana Univ. Math. J. 48 (1999), 899-936. · Zbl 0934.15023 |
| [10] | F. HIAI AND H. KOSAKI, Means of Hilbert space operators, Springer-Verlag, 2003. · Zbl 1048.47001 |
| [11] | F. HIAI, H. KOSAKI, D. PETZ AND B. RUSKAI, Families of completely positive maps associated with monotone metrics, Linear Algebra Appl., 48 (439) (2013), 1749-1791. · Zbl 1300.15009 |
| [12] | L. V. KANTOROVICH, Functional analysis and applied mathematics, Uspekhi Mat. Nauk, 3: 6 (28) (1948), 89-185, http://mi.mathnet.ru/eng/umn/v3/i6/p89. · Zbl 0034.21203 |
| [13] | H. KOSAKI, Positive definiteness and infinite divisibility of certain functions of hyperbolic cosine function, Internat. J. Math., 33 (7) (2022), 2250050. · Zbl 1496.42010 |
| [14] | H. KOSAKI, Positive definiteness of functions with applications to operator norm inequalities, Mem. Amer. Math. Soc., 212 (997), 2011. · Zbl 1227.47005 |
| [15] | H. KOSAKI, Strong monotonicity for various means, J. Func. Anal., 267 (2014), 1917-1958. · Zbl 1314.47023 |
| [16] | D. PETZ AND H. HASEGAWA, On the Riemannian metric of α -entropies of density matrices, Lett. Math. Phys., 38 (1996), 221-225. · Zbl 0855.58070 |
| [17] | W. SPECHT, Zur Theorie der elementaren Mittel, Math. Z, 74 (1960), 91-98, 10.1007/BF01180475. · Zbl 0095.03801 · doi:10.1007/BF01180475 |
| [18] | V. E. S. SZAB Ó, A class of matrix monotone functions, Linear Algebra Appl., 420 (2007), 79-85. (. · Zbl 1114.47022 |
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