For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup...For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup></em>, <em>a</em><sub><em>j </em></sub>≥ <em>a</em><sub><em>j</em>-1</sub>, <em>a</em><sub>0</sub> > 0, <em>j</em> = 1, 2, …, <em>n</em>, <em>a<sub>n</sub></em> > 0, a classical result of Enestr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>m-Kakeya says that all the zeros of <em>P</em> (<em>z</em>) lie in |<em>z</em>|≤ 1. This result was generalised by A. Joyall and G. Labelle, where they relaxed the non-negativity condition on coefficients. It was further generalized by M.A Shah by relaxing the monotonicity of some coefficients. In this paper, we use some known techniques and provide some more generalizations of the above results by giving more relaxation to the conditions.展开更多
文摘For the polynomial <em>P</em> (<em>z</em>) = <img src="Edit_94d094e0-dc15-4e21-b6cf-3fcb179d54b0.bmp" alt="" /><em>a<sub>j</sub>z<sup>j</sup></em>, <em>a</em><sub><em>j </em></sub>≥ <em>a</em><sub><em>j</em>-1</sub>, <em>a</em><sub>0</sub> > 0, <em>j</em> = 1, 2, …, <em>n</em>, <em>a<sub>n</sub></em> > 0, a classical result of Enestr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>m-Kakeya says that all the zeros of <em>P</em> (<em>z</em>) lie in |<em>z</em>|≤ 1. This result was generalised by A. Joyall and G. Labelle, where they relaxed the non-negativity condition on coefficients. It was further generalized by M.A Shah by relaxing the monotonicity of some coefficients. In this paper, we use some known techniques and provide some more generalizations of the above results by giving more relaxation to the conditions.
