We study some geometric properties associated with the t-geometric means A ♯_{t} B:= A^{1/2}(A^{-1/2}BA^{-1/2})^{t} A^{1/2}of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted., Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam., and Obsahuje seznam literatury
There are many relations involving the geometric means Gn(x) and power means [An(x γ )]1/γ for positive n-vectors x. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form (1 − λ)G γ n(x) + λAγ n(x) ≥ An(x γ ) and (1 − λ)G γ n(x) + λAγ n(x) ≤ An(x γ ) with parameters λ ∈ R and γ ∈ (0, 1). We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated.
The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.