What is it about?
In this paper, we investigate the Schur harmonic convexity for two classes of symmetric functions defined by Guan (2007), Xia and Chu (2009), and the Schur multiplicative convexity for a class of symmetric functions defined by Xia et al. (2010) by using a new method, generalize the main results of Xia et al. (2010). As applications, we establish some inequalities by use of the theory of majorization, in particular, we give some new geometric inequalities in the n-dimensional space.
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Why is it important?
(1) Our Theorem 4.3 is a generalization of the Theorem 4.5 in Xia et al. (2010). (2) The inequalities (4.9)-(4.12) can be found in Mitrinovic (1989), here we give a different proof of them. (3) The inequality (4.15) can be found in the Proposition 4 from Wu (2005), here we give a different proof of it.
Perspectives
For the two types of symmetric functions defined by Guan(2007), Xia, and Chu(2009), we expect readers to get Schur harmonic convexity more clearly. For a class of symmetric functions defined by Xia et al. (2010), we hope that readers will be able to get a clearer picture of Schur multiplicative convexity. Further more, we hope to use these results to generalize some new inequalities excitedly.
Xinping Li
Hunan Institute of Science and Technology
Read the Original
This page is a summary of: The Schur Multiplicative and Harmonic Convexities for Three Classes of Symmetric Functions, Journal of Function Spaces, December 2018, Hindawi Publishing Corporation,
DOI: 10.1155/2018/4036942.
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