On the Location of the Zeros of Certain Polynomials

S. D. Bairagi ; Vinay Kumar Jain ; T. K. Mishra ; L. Saha

Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 287 .

Résumé

We extend Aziz and Mohammad's result that the zeros, of a polynomial $P(z)=\sum_{j=0}^na_jz^j$, $ta_j\geq a_{j-1}>0$, $j=2,3,\dots,n$ for certain $t$ (${}>0$), with moduli greater than $t(n-1)/n$ are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial $P(z)$, of degree $n$, with complex coefficients, does not vanish in the disc \[ |z-a e^{ilpha}|/(2n);a>0,\max_{|z|=a}|P(z)|=|P(ae^{ilpha})|, \] for $r$ being the greatest positive root of the equation \[ x^n-2x^{n-1}+1=0, \] and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).

Détail

DOI : 10.2298/PIM1613287B

Classification : 30C15 30C10
Keywords: simple zeros, zero free region, refinement, upper bound for moduli of all zeros