The Integral Version of Popoviciu’s Inequality
نویسنده
چکیده
T. Popoviciu [7] has proved in 1965 an interesting characterization of the convex functions of one real variable, relating the arithmetic mean of its values and the values taken at the barycenters of certain subfamilies of the given family of points. The aim of our paper is to prove an integral analogue. Most of the people think that passing from a discrete inequality to its integral counterpart is just a snap. In practice, things are not so simple, due to the particularities of discrete probability fields that may hide essential facts. We will discuss here the case of Popoviciu’s inequality, a notable discrete inequality, whose statement is as follows: THEOREM 1. (T. Popoviciu [7]) If f is a convex function defined on an interval I, then ∑ 1 i1<···<ip n (λi1 + · · ·+λip) f (λi1xi1 + · · ·+λipxip λi1 + · · ·+λip ) ( n−2 p−2 )[ n− p p−1 n ∑ i=1 λi f (xi)+ ( n ∑ i=1 λi ) f ( λ1x1 + · · ·+λnxn λ1 + · · ·+λn )] , (Pn,p) for all families x1, ...,xn ∈ I, λ1, ...,λn ∈ [0,∞), n 3, and all integers p ∈ {2, ...,n− 1} . Actually, the inequality (P3,2) is equivalent to the property of convexity, and a simple argument based on mathematical induction yields the implication (P3,2) ⇒ (Pn,p) for all n ∈ N, n 3, and all p ∈ {2, ...,n−1}. Thus the essence of Theorem 1 is the connection between convexity and (P3,2). Popoviciu’s inequality has received a great deal of attention since its discovery in 1965, and appears in a series of monographs such as [2], [3], and [5]. See also [1] (for a higher dimensional analogue) and [4] (for some refinements). Mathematics subject classification (2000): Primary 26A51; Secondary 26D15.
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تاریخ انتشار 2009