- Research Article
- Open Access
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.
- Convex Function
- Integral Form
- Extended Form
- Concave Function
- Global Bound
Throughout this paper represents a finite sequence of real numbers belonging to a fixed closed interval , and is a positive weight sequence associated with .
There are many important inequalities which are particular cases of Jensen's inequality among which are the weighted inequality, Cauchy's inequality, the Ky Fan and Hölder's inequalities.
One can see that the lower bound zero is of global nature since it does not depend on but only on and the interval whereupon is convex.
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.
Let be a (strictly) positive, twice continuously differentiable function on , and . One has that
Both estimates are independent of .
The next assertion shows that (resp., ) exists and is unique.
Of particular importance is the following theorem.
The expression represents the best possible global upper bound for Jensen's inequality written in the form (1.4).
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.
holds for arbitrary and .
In particular, for we obtain that as required.
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 .
Taking , after an easy calculation it follows that . Therefore we consequently obtain the result.
holds for an arbitrary weight sequence .
This is the extended form of Schweitzer inequality.
For we get that the maximum of is attained at the point .
Hence, we have the following.
holds for an arbitrary weight sequence .
As a special case of the above inequality, that is, by putting and noting that imply , we obtain a converse of the well-known Cauchy's inequality.
In this form the Cauchy's inequality was stated in [2, page 80].
Note that the same result can be obtained from inequality (4.4) by taking .
Let . Since in this case is a concave function, applying the second part of Theorem 2.1, we get the following.
independently of .
and, letting , the converse of generalized inequality arises.
Note that the right-hand side of (4.9) is exactly the Specht ratio (cf. ).
and taking the limit , one has the following.
Finally, putting in (4.7) , we obtain the converse of discrete Hölder's inequality.
It is interesting to compare (4.12) with the converse of Hölder's inequality for integral forms (cf. ).
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