Generalizations of Popoviciu’s inequality
نویسنده
چکیده
We establish a general criterion for inequalities of the kind convex combination of f (x1) , f (x2) , ..., f (xn) and f (some weighted mean of x1, x2, ..., xn) ≥ convex combination of f (some other weighted means of x1, x2, ..., xn) , where f is a convex function on an interval I ⊆ R containing the reals x1, x2, ..., xn, to hold. Here, the left hand side contains only one weighted mean, while the right hand side may contain as many as possible, as long as there are finitely many. The weighted mean on the left hand side must have positive weights, while those on the right hand side must have nonnegative weights. This criterion entails Vasile Cı̂rtoaje’s generalization of the Popoviciu inequality (in its standard and in its weighted forms) as well as a cyclic inequality that sharpens another result by Vasile Cı̂rtoaje. The latter cyclic inequality (in its non-weighted form) states that 2 n ∑ i=1 f (xi) + n (n − 2) f (x) ≥ n n ∑ s=1 f ( x + xs − xs+r n ) , where indices are cyclic modulo n, and x = x1 + x2 + ... + xn n . This is the standard version of this note. A ”formal” version with more detailed proofs can be found at http://www.stud.uni-muenchen.de/~darij.grinberg/PopoviciuFormal.pdf However, due to these details, it is longer and much more troublesome to read, so it should be used merely as a resort in case you do not understand the proofs in this standard version.
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