On a variant of Čebyšev’s inequality of the Mercer type

We consider the discrete Jensen–Mercer inequality and Čebyšev’s inequality of the Mercer type. We establish bounds for Čebyšev’s functional of the Mercer type and bounds for the Jensen–Mercer functional in terms of the discrete Ostrowski inequality. Consequentially, we obtain new refinements of the considered inequalities.


Introduction
Let n ≥ 2 and let w = (w 1 , . . . , w n ) be a real n-tuple such that 0 ≤ W k = k i=1 w i ≤ W n , k = 1, . . . , n, W n > 0. (1) In [5] the following Čebyšev's inequality of the Mercer type: was proved for any real n-tuples x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) monotone in the same direction and real numbers a, b, c, d such that If x and y are monotonic in the opposite directions, inequality (2) is reversed.
Here, to be more precise, we cite that result with the slightly different notation.
In the same paper, the authors considered Čebyšev's functional (or Čebyšev's difference) of the Mercer type defined as the difference of the right-and left-hand sides of inequality (2). They established bounds in terms of the discrete Ostrowski inequality. Here we give more accurate bounds, which also provide refinements of inequality (2). In addition, using these results, we establish Ostrowski-like bounds for the Jensen-Mercer functional and, consequentially, a refinement of the Jensen-Mercer inequality.
Here, and in the rest of the paper, we assume l j=k x j = 0 when k > l.
Lemma 1 Let n ≥ 2 and let w be a real n-tuple such that (1) is fulfilled. Then for any real n-tuples x, y and real numbers a, b, c, d satisfying (3), the identity Proof For m = n + 2, we define m-tuples p, ξ , and ζ as Since w satisfies (1) it follows that Hence, we can apply identity (5). Its left-hand side is It can be easily seen that hence, on the right-hand side of (5) we have Calculating separately summands for i = 1 and i = m -1, we obtain Therefore, which is equal to the right-hand side of (6).
Using identity (6) and imposing stricter conditions than (3), we obtain refinements of inequality (2) which are more accurate than those previously established in [5].
Theorem 1 Let n ≥ 2 and let w be a real n-tuple such that (1) is fulfilled. Let x, y be real n-tuples monotonic in the same direction. Suppose that real numbers a, b, c, d and nonnegative real numbers r, s satisfy Then If x and y are monotonic in the opposite directions, then the inequalities in (10) are reversed and the term rs appears with the negative sign.
Proof Under the given assumptions, using identity (6), we obtain we obtain the first inequality in (10). Since r, s are nonnegative real numbers and obviously the second inequality in (10) immediately follows.
Remark 1 If in Theorem 2 we add assumption that R, S are nonnegative real numbers such that then we obtain refinements of the two inequalities proved in [5] under the same assumption. Namely, we have inequalities and, as a special case when w i = 1 (i = 1, . . . , n), we have inequalities 12 .

Bounds for the Jensen-Mercer functional
Jensen-Mercer inequality for a convex function f : (α, β) → R, real n-tuple x ∈ [a, b] n , and positive real n-tuple w, where -∞ ≤ α < a < b < β ≤ ∞, was proved in [6]. In [1], it was proved that it remains valid when x is monotonic and w satisfies conditions (1). Using our results from the previous section, we establish Ostrowski-like bounds for the Jensen-Mercer functional, i.e., the difference of the right-and left-hand sides of inequality (15).