On Some Inequalities Involving Harmonic Mean and Moments

Abstact: We derive bounds on the second order moment of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered and it is shown that the present bounds provide refinements of the bounds which exist in literature. As an application we obtain a lower bound for the spread of a positive definite matrix A in terms of traces of A, A-1 and A2. Our results compare favourably with those obtained by Wolkowicz and Styan (Bounds for eigenvalues using traces, Lin. Alg. Appl. 29, 471-506, 1980).


INTRODUCTION
Let a random variable x, discrete or continuous takes values in the interval a ≤ x ≤ b. Let H, µ ' 1 and µ ' 2 respectively denote the harmonic mean, arithmetic mean and second order moment about origin of the random variable x. The well-known Kantorovich inequality says that [1]  Sharma [3] shows that where S 2 is the variance of the random variable x. The inequality (1.3) provides a refinement of (1.1). Several authors have worked on such inequalities, their further refinements, extensions and applications. In particular, Krasnosel'skii and Krein [4] proved that Proof for any real number α, , ,..., i n = 1 2 . This gives , ,..., . Multiplying both sides of (2.3) by p i and adding these n inequalities, we get on simplification, The inequality (2.4) holds for all real values of α and therefore its discriminant must be non positive. Hence, we must have . (2.5) The inequality (2.1) now follows easily from (2.5). On using similar arguments we find that The inequality (2.2) now follows from (2.6); the discriminant of the quadratic equation in (2.6) must be non positive. Consider the system of linear equations p p p x p x p x p x p x p x p From (2.7), (2.8) and (2.9), we have On substituting the values of p 1 , p 2 and p 3 respectively from (2.10), (2.11) and (2.12) in we get the following cubic equation in variable x 2 , 14) It is easily seen that the root

Theorem-2.2
The inequalities (2.1) and (2.2) also hold good when H, µ 1 ' and µ 2 ' are respectively harmonic mean, arithmetic mean and second order moment about origin of a continuous random variable x whose probability density function φ x ( ) takes non-zero values in the interval [a,b], such that a<H<b, and a>0.

Proof:
for any real number α.This gives Multiplying both sides the inequality (2.22) by probability density function φ x ( ), we get on using the properties of the definite integral, This shows that inequality (2.1) remains valid when x is a continuous random variable. On using similar arguments we can show that (2.2) also hold good for the case when x is a continuous random variable.

Corollary-2.1
For a random variable which is discrete or continuous and takes values in the interval [a,b], we have The inequality (2.24) provides the refinement of the inequality (1.5).

Corollary-2.2
For a discrete or continuous random variable varying over the interval [a,b], we have . (2.28)

Corollary-2.3
For a random variable which is discrete or continuous and takes values in the interval [a,b], we must have µ µ The inequality (2.29) provides a refinement of the Krasnosel'skii and Krein inequality (1.4).

AN APPLICATION
Let λ λ λ 1 2 , ,..., , n be the eigenvalues of an n×n matrix A. The spread of A is defined by This quantity of Hermition and positive definite matrices is important in matrix analysis, and has applications in combinatorial optimization problems [5]. Several authors have given bounds for the spread. In particular, Wolkowicz and Styan [6], have obtained the following lower bound for the spread of positive definite matrix in terms of traces of A and A², ,n even Here we obtain a lower bound for the spread of positive definite matrix in terms of traces of A, A -1 and A 2 . Our results compare favourably then those obtained by Wolkowicz and Styan [6], (Example 1 and 2, below).

Theorem-3.1
Let A be n×n positive definite matrix with eigenvalues 0 . .