On location of zeros and polar derivatives of polynomials.

摘要

If pz=v=0 navzv is a polynomial of degree at most n, where z is a complex variable, then according to a famous inequality known as Bernstein's inequality, max z=1 p'z ≤nmaxz =1pz . 1 Equality is attained in (1) if p( z) has all its zeros at the origin.;If we restrict to the class of polynomials having all zeros in | z| ≥ 1, P. Erdos conjectured, and P. D. Lax proved that maxz =1p'z ≤n2 max z=1p z. 2 The inequality (2) is also sharp and equality holds if p(z) has all its zeros on |z| = 1.;For polynomials having all zeros in |z| ≥ 1, P. Turin proved that maxz =1p'z ≥n2 max z=1p z. 3 Again, the result is best possible and equality holds in (3) for any polynomial which has all its zeros on |z| = 1.;Over the years, these inequalities have been sharpened and generalized in several directions, including for polynomials having all zeros inside or outside |z| = K, K > 0.;We sharpen and generalize some of these results to polar derivatives (which generalize the ordinary derivative). We also obtain some analogous Ldelta inequalities for self-inversive and self-reciprocal polynomials.;Again, if pz=v=0 navzv is a polynomial of degree n, then by the Fundamental Theorem of Algebra, p(z) has n zeros in the complex plane. If p(z) has real, positive and monotonic coefficients, then according to a famous result called the Enestrom-Kakeya Theorem, all the zeros of p( z) lie in |z| ≤ 1. Though pretty, this theorem has a rather restrictive hypothesis which limits its scope and usefulness in applications. We obtain generalizations of this theorem by relaxing some of these restrictions. Our results include as special cases some of the known results in this direction.