arXiv:2111.03256v1 [math.FA] 5 Nov 2021 NEW INEQUALITIES FOR THE SPECTRAL GEOMETRIC MEAN HAMID REZA MORADI, SHIGERU FURUICHI AND MOHAMMAD SABABHEH Abstract. The main goal of this article is to present new inequalities for the spectral geometric mean A♮ t B of two positive definite operators A, B on a Hilbert space. The obtained results complement many known inequalities for the geometric mean A♯ t B. In particular, explicit comparisons between A♮ t B and A♯ t B are given, Ando-type inequalities are presented for A♮ t B and some other consequences. 1. Introduction Let B (H) be the C ∗ -algebra of all bounded linear operators on a Hilbert space H. If A ∈ B (H) is such that 〈Ax, x〉≥ 0 for all x ∈ C n , A is said to be positive semi-definite. If in addition A is invertible, it is positive definite. The class of positive definite matrices in M n will be denoted by B (H) + . The weighted geometric mean of A, B ∈B (H) + is defined by the equation A♯ t B = A 1 2  A − 1 2 BA − 1 2  t A 1 2 , 0 ≤ t ≤ 1. (1.1) When t = 1 2 , we simply write A♯B instead of A♯ 1 2 B. An interesting geometric meaning of A♯B is that it is a midpoint of A and B for a natural Finsler metric (Thompson’s part metric) on the cone of positive definite operators [16, 17]. The geometric mean ♯ t is a special case of operator means. Recall that an operator mean σ on B (H) + is a binary operation defined by AσB = A 1 2 f  A − 1 2 BA − 1 2  A 1 2 , where f : (0, ∞) → (0, ∞) is an operator monotone function, with f (1) = 1. Examples of other operator means are the weighted arithmetic and harmonic mean, defined respectively by A∇ t B = (1 − t)A + tB and A! t B = ( (1 − t)A −1 + tB −1 ) −1 , 0 ≤ t ≤ 1. 2010 Mathematics Subject Classification. Primary 47A64, 47A63. Secondary 47A50. Key words and phrases. Spectral geometric mean of positive operators, Ando inequality, Ando-Hiai inequality, Kantorovich constant. 1