Necessary and sucient conditions for symmetric homogeneous polynomial inequalities of degree four and six in real variables

Volume 5, Issue 5, pp 307--320 http://dx.doi.org/10.22436/jnsa.005.05.01

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Authors

Vasile Cirtoaje - Department of Automatic Control and Computers, University of Ploiesti, Ploiesti, Romania.

Abstract

Let \(f_n(x; y; z)\) be a symmetric homogeneous polynomial of degree \(n = 4\) or \(n = 6\), in three real variables. We give necessary and sufficient conditions to have \(f_n(x; y; z) \geq 0\) for all real numbers \(x; y; z\). Then, we apply the obtained results to prove several relevant symmetric homogeneous polynomial inequalities of degree four and six.

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Keywords

• Symmetric homogeneous polynomial
• necessary and sufficient conditions
• real variables.

MSC

•  26B25

References

• [1] T. Ando, Some Homogeneous Cyclic Inequalities of Three Variables of Degree Three and Four, Aust. J. Math. Anal. Appl., 7 (2010), 14 pages


• [2] V. Cirtoaje, On the Cyclic Homogeneous Polynomial Inequalities of Degree Four, Journal of Inequalities in Pure and Applied Mathematics, 10 (2009), 10 pages


• [3] V. Cirtoaje, Algebraic Inequalities , GIL Publishing House, (2006)


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