On region containing all the zeros of a polynomial

Eneström–Kakeya Theorem and Some of Its Generalizations

“…Govil and Rahman [30] also gave this type of result, and the same is stated as follows: Related results concerning the location of the zeros of a polynomial have also been presented by Aziz and Mohammad [7], Sun and Hsieh [61], Affane-Aji, Agarwal, and Govil [2], Affane-Aji, Biaz and Govil [3], Choo [15], Choo and Choi [17], Dalal and Govil [19], Gulzar [34,36], and Gilani [27]. The hypotheses of the following result, due to Jain [43] in 1988, are very much in the spirit of the Eneström-Kakeya theorem, although the conclusion involves the size of the real part of the zeros instead of the modulus: In the same paper, Jain gave a result by putting the monotonicity hypothesis on the real parts of the coefficients.…”

“…See [1,2,3,19] for several related results which apply to all polynomials with complex coefficients. In this survey, we explore the Eneström-Kakeya theorem and its generalizations.…”

On annulus containing all the zeros of a polynomial

“…The actual annulus containing all the zeros of P (z) is C 1 = {z : 0.8840 ≤ |z| ≤ 0.8899} with the area of 0.0328. According to [5]…”

“…As is expected, Theorem 2.1 gives a much better result than that of Corollary 1.1 with 67% improvement in the area of annulus.The actual annulus for this example is given by C 1 = {z : 0.7190 ≤ |z| ≤ 0.8801} with the area of 0.8093. As shown in[5], Corollary 1.2 gives the annulus C 2 = {z : 0.6147 ≤ |z| ≤ 1.1186} with the area of 2.7427. The annulus obtained from Theorem 2.1 is C 3 = {z : 0.6805 ≤ |z| ≤ 0.9048} with the area of 1.1172.…”

“…Recently Dalal and Govil [5] presented a unified result which contains all the above-mentioned approaches as special cases. More precisely they proved the following theorem.…”