Abstract
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.
Highlights
Our aim in this paper is to find the best possible global upper bound for 1.4
It is interesting to compare 4.12 with the converse of Holder’s inequality for integral forms cf. 4
Summary
Throughout this paper x {xi} represents a finite sequence of real numbers belonging to a fixed closed interval I a, b , a < b, and p {pi}, pi 1 is a positive weight sequence associated with x. If f is a convex function on I, the well-known Jensen’s inequality 1, 2 asserts that 0 ≤ pif xi − f pixi. One can see that the lower bound zero is of global nature since it does not depend on p, x but only on f and the interval I whereupon f is convex. We prove that 0 ≤ pif xi − f pixi ≤ Tf a, b ,. For a strictly positive convex function f, Jensen’s inequality can be stated in the form 1 ≤ pif xi . It is not difficult to prove that 1 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 obtain converses of some important inequalities
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