REFINEMENTS OF JENSEN-MERCER’S INEQUALITY FOR INDEX SET FUNCTIONS WITH APPLICATIONS

. Some reﬁnements of Jensen-Mercer’s inequality are presented. They are used to reﬁne few inequalities among various means of Mercer’s type, and they are further generalized for linear functionals.


INTRODUCTION
In paper [3] A. McD.Mercer proved the following variant of Jensen's inequality, to which we will refer as to the "Jensen-Mercer's inequality".
In this paper we give some refinements of (1) and we present several applications of them.In Section 2 we first prove the Jensen-Mercer's inequality for weights satisfying conditions as for the reversed Jensen's inequality (see for example [4, p. 83]), and after that we prove refinements of Theorem A, using an index set function.In Section 3 we use these results to refine some well known inequalities among arithmetic, geometric, harmonic, power and quasi-arithmetic means of Mercer's type.In Section 4 we generalize our main results for linear isotonic functionals.
Let To prove Theorem 2.1, we need the following Lemma: Proof of Theorem 2.1.Weights w 1 , . . ., w n satisfy conditions (2) and , so by Lemma 2.2 and by the reversed Jensen's inequality, we have Let I be a finite nonempty set of positive integers, and let f : where W I = i∈I w i .If we define the index set function F as then the following theorem is valid.
Let I and J be finite nonempty sets of positive integers such that If W I • W J < 0, then the inequality (3) is reversed.
Proof.Since f is convex, the same is also true for the function g : [a, b] → R defined as g(y) = f (a + b − y), y ∈ [a, b].Hence, the following inequality holds for every y 1 , y 2 ∈ [a, b] and u 1 , u 2 > 0 (4) , i.e., (5) is reversed.This is a simple consequence of (4) after we make the substitutions u 1 +u 2 , and y 2 → y 2 (similarly as in the proof of the reversed Jensen's inequality).
Suppose that W I > 0 and W J > 0. If we let in (5), then we obtain Multiplying the above inequality by (−1) and adding to the both sides the term it follows that In case when W I • W J < 0, for instance W I > 0 and W J < 0, we again let and reversed (3) follows from reversed (5).

. , I k be finite nonempty sets of positive integers such that
If W I 1 > 0 and W I j < 0 (j = 2, . . ., k), then the inequality (6) is reversed.
Proof.Directly from Theorem 2.3 by induction.
The following corollaries give refinements of Theorem A.
Proof.Suppose that w i ≥ 0 for i = 2, . . ., n.First we show that and since w k ≥ 0, it follows that F ({k}) ≥ 0. Now, by Theorem 2.3, for all k ∈ {2, . . ., n}.Suppose that w i ≤ 0 for i = 2, . . ., n, W In > 0 and and then add to the both sides −w n x n , we obtain Multiplying the above inequality by 1 Since, wn
The inequality (10) can be proved in the similar way.
Remark 2.7.Analogous assertions can be formulated for concave functions using the fact that f is concave iff −f is convex.

APPLICATIONS
Let A n , G n , H n , and M [r] n be the arithmetic, geometric, harmonic, and power mean of order r, respectively, of the real numbers x i ∈ [a, b], where 0 < a < b, formed with the positive weights w i (i = 1, . . ., n).For the various properties of these means and relations among them we refer the reader to [2].For example, it is well known that If we define r , then we have the following results.
(ii) Applying Corollary 2.5 to the convex function f (x) = exp x, and replacing a, b, and x i with ln a, ln b, and ln x i respectively, we obtain W n 1 Proof.Directly from Theorem 3.1 by the substitutions a For r ≥ 1, the inequalities (13) are reversed.
Proof.Suppose that r ≤ 1. Applying Corollary 2.5 to the convex function f (x) = x 1 r , and replacing a, b, and x i with a r , b r , and x r i respectively, we obtain (13) since in this case r is concave, so the inequalities (13) are reversed.
Remark 3.5.Obviously, the assertion (ii) from Theorem 3.1 is also direct consequence of Theorem 3.3.
Proof.Suppose that s > 0. Applying Corollary 2.5 to the convex function f (x) = x s r , and replacing a, b, and x i with a r , b r , and x r i respectively, we obtain (14) since If s < 0, then the function f (x) = x s r is concave, so ( 14) is reversed.
Let ϕ : [a, b] → R be a strictly monotonic and continuous function, where is well defined and is called quasi-arithmetic mean of x with weights w (see for example [2, p. 215]).If we define then we have the following results.
Remark 3.10.Analogous assertions can be formulated for the means of Mercer's type formed with weights satisfying (2).

FURTHER GENERALIZATION
Let E be a nonempty set, A be an algebra of subsets of E, and L be a linear class of real valued functions f : E → R having the properties: Let A : L → R be an isotonic linear functional having the properties: is an isotonic linear functional with A 1 (1) = 1.Furthermore, we observe that (18) In [1], under the above assumptions, the following variant of the Jessen's inequality Theorem 4.1.Under the above assumptions, if ϕ is convex, then Proof.Since ϕ is continuous and convex, the same is also true for the function A(χ E 1 ) , and A(χ E\E 1 ) , then p + q = 1 by the equality (18), and pt 1 + qt 2 = A(f ) by the equality (19).Similarly, we can use that Hence, we have Multiplying the above inequality with (−1) and adding to the both sides the term ϕ (a) + ϕ (b) − A(ϕ (f )) we obtain the first inequality in (21).
To prove the remaining two inequalities in (21), observe that they are simple consequences of (20) applied to the isotonic linear functional A 1 defined for and Proof.Directly from Theorem 4.1 by induction.If ϕ is concave, then the inequalities (24) and (25), with max replaced by min, are reversed.
Theorem A. Let [a, b] be an interval in R, and x 1 , . . ., x n ∈ [a, b].Let w 1 , . . ., w n be nonnegative real numbers such that W n = n i=1 w i > 0. If f is a convex function on [a, b], then (1) [a, b]  be an interval in R, andx 1 , . . ., x n ∈ [a, b] such that 1 Wn n i=1 w i x i ∈ [a, b].If f is a convex function on [a, b], then (1) holds.