New lower bounds for arithmetic, geometric, harmonic mean inequalities and entropy upper bound

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.

Some More Inequalities for Arithmetic Mean, Harmonic Mean and Variance

We derive bounds on the variance of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered, and an operator version is obtained. Some refinements of the Kantorovich inequality are obtained. Bounds for the largest and smallest eigenvalues of a positive definite matrix are also obtained.

Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Upper and lower bounds for numerical radii of block shifts

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Singular Value and Arithmetic-geometric Mean Inequalities for Operators

A singular value inequality for sums and products of Hilbert space operators is given. This inequality generalizes several recent singular value inequalities, and includes that if A, B, and X are positive operators on a complex Hilbert space H, then sj ( A 1/2 XB 1/2 ) ≤ 1 2 ‖X‖ sj (A+B) , j = 1, 2, · · · , which is equivalent to sj ( A 1/2 XA 1/2 −B 1/2 XB 1/2 ) ≤ ‖X‖ sj (A⊕B) , j = 1, 2, · · ...