A Jensen-type inequality in the framework of 2-convex systems

Precupescu, George

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creative_2023_32_2_193_200

DOI:https://doi.org/10.37193/CMI.2023.02.07

Issue no: Vol 32/2023 no. 2 Tags: convex function, Majorization, 3-convex function, relative convexity, equal variable method

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Let $A_S$ be the solution set of the system $x_{1} + x_{2} + \ldots + x_{n} = ns$ , $e(x_{1}) + e(x_{2}) + \ldots + e(x_{n}) = nk$ , $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ , where $e:I \to \RR$ is a (fully extended) strictly convex or concave function. We call such a system 2–convex and prove the existence of two special points $\omega, \Omega \in A_S$ such that for all $x \in A_S$ and for all $f:I \to \RR$ strictly 3-convex with respect to $e$ , the following inequality holds: $\forall x \in A_S \Rightarrow E_f(\omega) \leq E_f(x) \leq E_f(\Omega)$ where $E_f(x) = f(x_1) + f(x_2) + \ldots + f(x_n)$ . This may be seen as a broader version of the equal variable method of V. C\^{i}rtoaje. It follows that $\omega$ and $\Omega$ have at most three distinct components and we also give a detailed analysis of their structure.