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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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[1] Alzer, H., A short proof of Ky Fan’s inequality, Arch. math., 27, 199-200, (1991) · Zbl 0759.26014
[2] Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer-Verlag Berlin · Zbl 0513.26003
[3] Gabler, S., Aufgabe 830, Elem. math., 3, 124-125, (1980)
[4] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Some simple inequalities satisfied by convex functions, Messenger math., 58, 145-152, (1929) · JFM 55.0740.04
[5] Kuang, K.J., Applied inequalities, (2004), Shangdong Science and Technology Press Jinan, (in Chinese)
[6] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic Press New York · Zbl 0437.26007
[7] McGregor, M.T., On some inequalities of Ky Fan and wang – wang, J. math. anal. appl., 180, 182-188, (1993) · Zbl 0803.26010
[8] Mercer, A.McD., A short proof of Ky Fan’s arithmetic – geometric inequality, J. math. anal. appl., 204, 940-942, (1996) · Zbl 0872.26007
[9] Niculescu, C.P., Convexity according to the geometric Mean, Math. inequal. appl., 2, 155-167, (2000) · Zbl 0952.26006
[10] Mitrinović, D.S., Analytic inequalities, (1970), Springer-Verlag New York · Zbl 0199.38101
[11] Pečarić, J.; Proschan, F.; Tong, Y.L., Convex functions, partial orderings, and statistical applications, (1992), Academic Press New York · Zbl 0749.26004
[12] Zhang, X.M., Geometrically-convex functions, (2004), Anhui Univ. Press Hefei, (in Chinese)
[13] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge Univ. Press New York · Zbl 0729.15001
[14] Zhang, F.Z., Matrix inequalities by means of block matrices, Math. inequal. appl., 4, 481-490, (2001) · Zbl 1012.15013
[15] Mitrinović, D.S.; Pečarić, J.E.; Volence, V., Recent advance in geometric inequalities, (1989), Kluwer Academic Publishers · Zbl 0679.51004
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