On Inequalities Involving Moments of Discrete Uniform Distributions

Some inequalities for the moments of discrete uniform distributions are obtained. The inequalities for the ratio and difference of moments are given. The special cases give the inequalities for the standard power means.


INTRODUCTION
L et x 1 , x 2, … , x n denote n real numbers such that a ≤ x i ≤ b, i = 1,2,…,n. The r th order moment µ r / of these numbers is defined as It may be noted that M -1 , M 0 and M 1 respectively define harmonic mean, geometric mean and Arithmetic mean. It is well known that the power mean M r is an increasing function of r. For 0 < a ≤ x i ≤ b, i = 1, 2,..., n, we have, [1], See also [2]. Some bounds for the difference and ratio of moments have also been investigated in literature; see [3][4][5][6].
Our main results give the refinements of the inequalities (1.4) and (1.5) when the minimum and maximum values of x 1 namely, a and b, and the value of n is prescribed, (Theorem 2.1 and 2.2, below). The bounds for the difference and ratio of moments are obtained (Theorem 2.3-2.6, below). We also discuss the cases when the inequalities reduce to equalities. As the special cases, we get various bounds connecting lower order moments and the standard means of the n real numbers, (Inequalities 3.1 -3.33, below), also see [7][8][9].

MAIN RESULTS
where n ≥ 3, r is a positive real number and is any non-zero real number such that r > s For s < r < 0 the reverse inequality holds. The inequality (2.1) becomes equality when The inequality (2.1) provides a refinement of the inequality (1.4).
Proof. The r th order moment of n real numbers x i , with x 1 = a and x n = b can be written as Substituting the value from (2.6) in (2.4), we immediately get (2.1). For s < r < 0, inequality (2.3) reverses its order [2]. It follows therefore that inequality (2.1) will also reverse its order for s < r < 0.
where n ≥ 3 and r is a non-zero real number. The inequality (2.7) becomes equality when (2.11) Substituting the value from (2.11) in (2.9) we immediately get (2.7).
where r is a positive real number and s is a nonzero real number such that r > s. For s < r < 0, the reverse inequality holds. For n = 2 inequality (2.13) becomes equality. For n ≥ 3, inequality (2.13) is sharp; equality holds when Proof. It follows from the inequality (2.1) that vanishes at The value of the second order derivative  .
where r is a positive real number and s is any non-zero real number such that r > s. For s < r < 0, the reverse inequality holds.
where r is a positive real number and s is a non zero real number such that r > s. We now find the minimum value of the right side expression in (2.24) as ′ µ s varies over the interval [a s ,b s ].
Consider a function On the other hand we conclude from theorem 2.1 that (2.30) Combining (2.25), (2.26) and (2.27), we find that for s < r < 0, where r is a non zero real number. We now find the minimum value of the right hand side expression in (2.33). Consider a function

SOME SPECIAL CASES
From the application point of view, it is of interest to know the bounds for the first four moments and the bounds for standard power means (namely, Arithmetic mean, ( µ 1 / = x ), Geometric mean (G) and Harmonic mean (H)). These bounds are also of fundamental interest in the theory of inequalities. By assigning particular values to r and s, in the generalized inequalities obtained in this paper, we can find inequalities connecting various power means and moments. The following inequalities can be deduced easily from the generalized inequalities given in Theorems 2.1 and 2.2: