On the derivative and maximum modulus of a polynomial
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If p(z) = nv=0 av zv is a polynomial of degree n, having all its zeros in |z| ≤ 1, then it was proved by Tur´an that | p (z)| ≥ (n/2)max|z|=1 | p(z)|. This result of Tur´an was generalized by Govil, who proved that if p(z) has all its zeros in |z| ≤ K, K ≥ 1, then max|z|=1 | p (z)| ≥ (n/(1 + K n ))max|z|=1 | p(z)|, K ≥ 1. In this paper, we sharpen this, and some other related results. Copyright © 2006 K. K. Dewan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and statement of results If p(z) =
n
v =0 a v z
v
is a polynomial of degree n, then it is well known that
max p (z) ≤ nmax p(z). |z|=1
|z|=1
(1.1)
The above inequality, which is an immediate consequence of Bernstein’s inequality on the derivative of a trigonometric polynomial, is best possible with equality holding for the polynomial p(z) = λzn , λ being a complex number. If we restrict ourselves to the class of polynomials having no zeros in |z| < 1, then the above inequality can be sharpened. In fact Erd¨os conjectured and later Lax [7] proved that if p(z) = 0 in |z| < 1, then
max p (z) ≤ |z|=1
n max p(z). 2 |z|=1
(1.2)
If the polynomial p(z) of degree n has all its zeros in |z| ≤ 1, then it was proved by Tur´an [9], that
max p (z) ≥ |z|=1
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 54816, Pages 1–6 DOI 10.1155/JIA/2006/54816
n max p(z). 2 |z|=1
(1.3)
2
On derivative and maximum modulus of a polynomial
The inequalities (1.2) and (1.3) are also best possible, and become equality for polynomials which have all its zeros on |z| = 1. The above inequality (1.3) of Tur´an [9] was generalized by Govil [3], who proved that if p(z) is a polynomial of degree n having all its zeros in |z| ≤ K, then
n max p(z), 1 + K |z|=1
n max p(z), n 1 + K |z|=1
max p (z) ≥ |z|=1
max p (z) ≥ |z|=1
if K ≤ 1,
(1.4)
if K ≥ 1.
(1.5)
Both the above inequalities are best possible, with equality in (1.4) holding for p(z) = (z + K)n , while in (1.5) the equality holds for the polynomial p(z) = zn + K n . The inequality (1.4) was also proved by Malik [8]. The inequality (1.5) was later sharpened by Govil [4, page 67], who proved the following theorem. n
v =0 a v z
Theorem 1.1. If p(z) = in |z| ≤ K, K ≥ 1, then
v,
an = 0, is a polynomial of degree n having all its zeros
n max p(z) n 1 + K |z|=1
max p (z) ≥ |z|=1
n a n −1 K n − 1 K n −2 − 1 1 + − + a1 1 − 2 n n n−2 K K 1+K
(1.6)
if n > 2, and
max p (z) ≥ |z|=1
n n p(z) + K − 1 a1 max 1 + K n |z|=1 Kn + 1
(1.7)
if n = 2. The above inequalities are best possible and are attained for the polynomial p(z) = zn + n K . In this paper, we prove the following refinement of Theorem 1.1, which in turn gives the ref
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