Abstract
It is proved that if \(\lambda\in\mathbb{R}^{n}\), \(1<\lambda_{n}\leq\lambda_{n-1}\leq\dots\leq\lambda_{1}\), \(\Re:=\{\nu\in\mathbb{R}_{+}^{n},(\lambda,\nu)\leq 1\}\), \(\mathcal{M}:=\{\nu\in\mathbb{R}_{+}^{n},\sum_{j=1}^{n-1}\lambda_{j}\nu_{j}+\lambda_{n-1}\nu_{n}\leq 1\}\) and the polynomial \(P(\xi)=P(\xi_{1},\dots,\xi_{n})\) is \(\Re\)-hyperbolic with respect to the vector \(\eta=(\eta_{1},\dots,\eta_{n})\in R^{n}\), \(\eta_{n}\neq 0\), then it is also \(\mathcal{M}\)-hyperbolic with respect to \(\eta\).
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The work is supported by the R&D Program of the Russian-Armenian University.
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Margaryan, V.N., Ghazaryan, H.G. On Certain Class of Weighted Hyperbolic Polynomials. J. Contemp. Mathemat. Anal. 56, 319–331 (2021). https://doi.org/10.3103/S1068362321060066
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DOI: https://doi.org/10.3103/S1068362321060066