ON THE LOCATION OF THE ZEROS OF CERTAIN POLYNOMIALS

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

Abstract: 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^{i\alpha}|<a/(2n);a>0,\max_{|z|=a}|P(z)|=|P(ae^{i\alpha})|,
for $r<a<2,r$ 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).

Keywords: simple zeros; zero free region; refinement; upper bound for moduli of all zeros

Classification (MSC2000): 30C15; 30C10

Full text of the article: (for faster download, first choose a mirror)

• Compressed DVI file (20 kilobytes)
• Compressed PostScript file (404 kilobytes)
• PDF file (140 kilobytes)