On a Converse of Jensen's Discrete Inequality
© Slavko Simic. 2009
Received: 10 July 2009
Accepted: 6 December 2009
Published: 15 December 2009
We give the best possible global bounds for a form of discrete Jensen's inequality. By some examples the fruitfulness of this result is shown.
It is not difficult to prove that is the best possible global lower bound for Jensen's inequality written in the above form. Our aim in this paper is to find the best possible global upper bound for (1.4). We will show with examples that by following this approach one may consequently obtain converses of some important inequalities.
Our main result is contained in what follows.
Of particular importance is the following theorem.
We will give proofs of the previous assertions related to the first part of Theorem 2.1. Proofs concerning concave functions go along the same lines.
Proof of Theorem 2.1.
We apply the method already shown in . Namely, since , there is a sequence such that .
Proof of Theorem 2.2.
Since is strictly convex on and , we conclude that is monotone decreasing on with . Since is continuous, there exists unique such that . Also , showing that is attained at the point . The proof is completed.
Proof of Theorem 2.3.
In the sequel we will give some examples to demonstrate the fruitfulness of the assertions from Theorem 2.1. Since all bounds will be given as a combination of means from the Stolarsky class, here is its definition.
and this form is introduced by Stolarsky in .
Most of the classical two variable means are special cases of the class . For example, is the arithmetic mean , is the geometric mean , is the logarithmic mean , is the identric mean , and so forth. More generally, the th power mean is equal to .
This is the extended form of Schweitzer inequality.
Hence, we have the following.
In this form the Cauchy's inequality was stated in [2, page 80].
Note that the right-hand side of (4.9) is exactly the Specht ratio (cf. ).
It is interesting to compare (4.12) with the converse of Hölder's inequality for integral forms (cf. ).
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