engleski

An improvement of the converse Jensen inequality

An improvement of the converse Jensen inequalityJ. Pečarić and J. Perić Let I be an interval in R and f:I→ℝ a convex function on I. If x=(x₁, …, x_{;n};) is any n-tuple in Iⁿ and p=(p₁, …, p_{;n};) a nonnegative n-tuple such that P_{;n};=∑_{;i=1};ⁿp_{;i};>0, then the well known Jensen's inequality f((1/(P_{;n};))∑_{;i=1};ⁿp_{;i};x_{;i};)≤(1/(P_{;n};))∑_{;i=1};ⁿp_{;i};f(x_{;i};) <label>jen</label> holds. If f is strictly convex then (<ref>jen</ref>) is strict unless x_{;i};=c for all i∈{;j:p_{;j};>0};. Jensen's inequality is probably the most important of all inequalities: it has many applications in mathematics and statistics and some other well known inequalities are its special cases (such as Cauchy's inequality, Hölder's inequality, A-G-H inequalities, etc.). Strongly related to Jensen's inequality is so called the converse Jensen inequality (1/(P_{;n};))∑_{;i=1};ⁿp_{;i};f(x_{;i};)≤((M-x)/(M-m))f(m)+((x-m)/(M-m))f(M), <label>cjen</label> which holds when f:I→ℝ is a convex function on I, [m, M]⊂I, -∞<m<M<+∞, p, x are as in (<ref>jen</ref>) and x=(1/P_{;n};)∑_{;i=1};ⁿp_{;i};x_{;i};. If f is strictly convex then (<ref>cjen</ref>) is strict unless x_{;i};∈{;m, M}; for all i∈{;j:p_{;j};>0};. Here we give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.

Converse Jensen's inequality; Converse Jessen's inequality

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