Improved logarithmic-geometric mean inequality and its application

Document Type : Research Paper


School of Mathematics and Statistics‎, ‎Chongqing Three Gorges University‎, ‎Chongqing‎, ‎404100‎, ‎P.R‎. ‎China.


In this short note, we present a refinement of the logarithmic-geometric mean inequality. As an application of our result, we obtain an operator inequality associated with geometric and logarithmic means.


Main Subjects

R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl. 413 (2006), no. 2-3, 355--363.
R. Bhatia, Positive Definite Matrices, Princeton Univ. Press, Princeton, 2007.
R. Bhatia and P. Grover, Norm inequalities related to the matrix geometric mean, Linear Algebra Appl. 473 (2012), no. 2, 726--733.
D. Drissi, Sharp inequalities for some operator means, SIAM J. Matrix Anal. Appl. 28 (2006), no. 3, 822--828.
J.I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989), no. 3, 341--348.
J.I. Fujii, Y. Seo and T. Yamazaki, Norm inequalities for matrix geometric means of positive definite matrices, Linear Multilinear Algebra 64 (2016), no. 3, 512--526.
S. Furuichi, K. Yanagi and K. Kuriyama, A note on operator inequalities of Tsallis relative operator entropy, Linear Algebra Appl. 407 (2005), no. 1, 19--31.
T. Furuta, Invitation to Linear Operators, Taylor & Francis, London-New York, 2001.
T. Furuta, Reverse inequalities involving two relative operator entropies and two relative entropies, Linear Algebra Appl. 403 (2005), no. 1, 24--30.
T. Furuta, Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications, Linear Algebra Appl. 412 (2006), no. 2-3, 526--537.
K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005), no. 1, 109--118.
L. Zou, Operator inequalities associated with Tsallis relative operator entropy, Math. Inequal. Appl. 18 (2015), no. 2, 401--406.