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.



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 Precupescu, George