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On the generalised sum of squared logarithms inequality. (English) Zbl 1312.26032

Summary: Assume \(n\geq2\). Consider the elementary symmetric polynomials \(e_{k}(y_{1},y_{2},\dots, y_{n})\) and denote by \(E_{0},E_{1},\dots,E_{n-1}\) the elementary symmetric polynomials in reverse order \(E_{k}(y_{1},y_{2},\dots,y_{n}):=e_{n-k}(y_{1},y_{2},\dots,y_{n})= \sum_{i_{1}<\cdots<i_{n-k}} y_{i_{1}}y_{i_{2}}\cdots y_{i_{n-k}}\), \(k\in\{0,1,\dots,n-1 \}\). Let, moreover, \(S\) be a nonempty subset of \(\{0,1,\dots,n-1\}\). We investigate necessary and sufficient conditions on the function \(f:I\to\mathbb{R}\), where \(I\subset\mathbb{R}\) is an interval, such that the inequality \[ f(a_{1})+f(a_{2})+\cdots+f(a_{n})\leq f(b_{1})+f(b_{2})+\cdots+f(b_{n})\qquad\qquad (\ast) \]
holds for all \(a=(a_{1},a_{2},\dots,a_{n})\in I^{n}\) and \(b=(b_{1},b_{2},\dots,b_{n})\in I^{n}\) satisfying \(E_{k}(a)< E_{k}(b)\) for \(k\in S\) and \(E_{k}(a)=E_{k}(b)\) for \(k\in\{0,1,\dots,n-1\}\setminus S\). As a corollary, we obtain our inequality \((\ast)\) if \(2\leq n\leq4\), \(f(x)=\log^{2}x\) and \(S=\{1,\dots,n-1\}\), which is the sum of squared logarithms inequality previously known for \(2\leq n\leq3\).

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
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References:

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