Turán-Type inequalities for polynomials with restricted Zeros -II

Annular region containing all the zeros of lacunary-type polynomials

“…For this reason, there is always a need for better estimates for the region containing some or all the zeros of a polynomial. A review on the location of zeros of polynomials can be found in [4,9,13,16]. One of the most elegant results on the bounds of zeros of a polynomial with restrictions on its coefficients, known as Eneström-Kakeya theorem (for reference see section 8.3 of [18]), states that if P (z) = n j=0 a j z j is a polynomial of degree n with real coefficients, such that a n ≥ a n−1 ≥ ... ≥ a 1 ≥ a 0 > 0, then P (z) has all its zeros in |z| ≤ 1.…”

“…One of the most elegant results on the bounds of zeros of a polynomial with restrictions on its coefficients, known as Eneström-Kakeya theorem (for reference see section 8.3 of [18]), states that if P (z) = n j=0 a j z j is a polynomial of degree n with real coefficients, such that a n ≥ a n−1 ≥ ... ≥ a 1 ≥ a 0 > 0, then P (z) has all its zeros in |z| ≤ 1. In the literature, there exist various extensions and generalizations of Eneström-Kakeya theorem [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In 1996, Aziz and Zargar [1] proved the following results for the regions containing the zeros of the lacunary-type polynomials.…”