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Some properties of a class of symmetric functions. (English) Zbl 1127.26018
This paper considers the Schur convexity of the functions
${\mathcal F}_n^k(x) = \sum_{1\leq i_1\leq \cdots \leq i_k\leq n}\prod_{j=1}^k{x_{i_j}\over (1-x_{i_j})}$
where the $$(x_1, \dots x_n)$$ lie in some subset of $$D=[0,1[^n$$. The main results are: (i) $${\mathcal F}_n^1, n\geq2$$, is strictly Schur convex on $$D$$; (ii) $${\mathcal F}_n^2, n\geq 3$$, is strictly Schur convex on $$B$$ the subset of $$D$$, where $$\sum_{i=1}^n x_i\leq 1$$; (iii) $${\mathcal F}_n^{n-1}, n\geq 4$$, is strictly Schur concave on the subset of $$C$$ of $$B$$ where $$x_i>0, 1\leq i\leq n$$; (iv) $${\mathcal F}_n^n, n\geq 3$$, is strictly Schur concave on $$]0, 1/2]^n$$ and also on $$C$$. Using a recent definition of X. M. Zhang [ Anhui Univ. Press (2004)], analogous to the definition of geometrical convexity, the author also proves that all the functions $${\mathcal F}_n^k$$ are Schur geometrically convex on $$]0,1[^n$$. Once these properties have been proved, using the differential criteria for Schur convexity and geometric Schur convexity, it is possible to give very simple proofs of the Shapiro and Ky Fan inequalities. Several other applications are also given.

##### MSC:
 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations
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