Functions and Analysis

    

Majorization in the Framework of 2-Convex Systems

Authors: George Precupescu

We define a 2-convex system by the restrictions $x_{1} + x_{2} + \ldots + x_{n} = ns$, $e(x_{1}) + e(x_{2}) + \ldots + e(x_{n}) = nk$, $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ where $e:I \to \RR$ is a strictly convex function. We study the variation intervals for $x_k$ and give a more general version of the Boyd-Hawkins inequalities. Next we define a majorization relation on $A_S$ by $x\preccurlyeq_p y$ $\Leftrightarrow$ $T_k(x) \leq T_k(y) \ \ \forall 1 \leq k \leq p-1$ and $B_k(x) \leq B_k(y) \ \ \forall p+2 \leq k \leq n$ (for fixed $1 \leq p \leq n-1$) where $T_k(x) = x_1 + \ldots + x_k$, $B_k(x) = x_k + \ldots + x_n$. The following Karamata type theorem is given: if $x, y \in A_S$ and $x\preccurlyeq_p y$ then $f(x_1) + f(x_2) + \ldots + f(x_n) \leq f(y_1) + f(y_2) + \ldots + f(y_n)$ $\forall$$f:I \to \RR$ 3-convex with respect to $e$. As a consequence, we get an extended version of the equal variable method of V. Cîrtoaje

Comments: 30 Pages. This is an English version (the original v1 was in Romanian)

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Submission history

[v1] 2020-06-23 10:49:42
[v2] 2020-06-26 07:48:37 (removed)
[v3] 2020-06-27 00:56:34 (removed)
[v4] 2020-07-04 13:01:43

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