Non-commutative arithmetic-geometric mean inequality

The matrix arithmetic-geometric mean inequality revisited

Non-commutative A-g Mean Inequality

In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality ab + (r − 1)b ≥ ra for two positive numbers a, b and r > 1. We show that under some assumptions the non-commutative analogue for ab which satisfies this inequality is unique and equal to r-mean. The case 0 < r < 1 is also considered. In particular, we give a new characterization of the geometric mean.

A Relationship between Subpermanents and the Arithmetic-Geometric Mean Inequality

Using the arithmetic-geometric mean inequality, we give bounds for k-subpermanents of nonnegative n × n matrices F. In the case k = n, we exhibit an n 2-set S whose arithmetic and geometric means constitute upper and lower bounds for per(F)/n!. We offer sharpened versions of these bounds when F has zero-valued entries.

An Arithmetic-Geometric-Harmonic Mean Inequality Involving Hadamard Products

Given matrices of the same size, A = a ij ] and B = b ij ], we deene their Hadamard Product to be A B = a ij b ij ]. We show that if x i > 0 and q p 0 then the n n matrices q j # are positive deenite and relate these facts to some matrix valued arithmetic-geometric-harmonic mean inequalities-some of which involve Hadamard products and others unitarily invariant norms. It is known that if A is p...

Best Upper Bounds Based on the Arithmetic-geometric Mean Inequality

In this paper we obtain a best upper bound for the ratio of the extreme values of positive numbers in terms of the arithmetic-geometric means ratio. This has immediate consequences for condition numbers of matrices and the standard deviation of equiprobable events. It also allows for a refinement of Schwarz’s vector inequality.