Generalizations of some Bernstein-type inequalities for the polar derivative of a polynomial
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neralizations of some Bernstein-type inequalities for the polar derivative of a polynomial Adil Hussain1
· Abrar Ahmad1
Received: 22 January 2020 / Accepted: 24 May 2020 © Università degli Studi di Ferrara 2020
Abstract In this paper, we shall generalize two recently proved results of Mir (Ann Univ Ferrara 65:327–336, 2019) concerning the polar derivative of a polynomial. Keywords Polar derivative · Bernstein inequality · Maximal modulus · Zeros Mathematics Subject Classification 30A10 · 30C10 · 30D15
1 Introduction Let Pn denote the class of all complex polynomials P(z) := nj=0 c j z j of degree n. A classical majorization result due to Bernstein is that, for two polynomials P and Q with Q ∈ Pn , deg P ≤ deg Q and Q(z) = 0 for |z| > 1, the majorization |P(z)| ≤ |Q(z)| on the unit circle |z| = 1 implies the majorization of their derivatives |P (z)| ≤ |Q (z)|. In particular, this majorization result allows to establish the famous Bernstein inequality [2] for the sup-norm on the unit circle: namely, if P ∈ Pn , then max P (z) ≤ n max P(z).
|z|=1
|z|=1
(1.1)
The above inequality (1.1) was proved by Bernstein in 1912. Later in 1985, Frappier et al. [4] strengthened (1.1), by proving that if P ∈ Pn , then ilπ max P (z) ≤ n max P e n .
|z|=1
B
1≤l≤2n
(1.2)
Adil Hussain [email protected] Abrar Ahmad [email protected]
1
Department of Mathematics, University of Kashmir, Srinagar 190006, India
123
ANNALI DELL’UNIVERSITA’ DI FERRARA
Clearly (1.2) represents a refinement of (1.1), since the maximum of |P(z)| on |z| = 1 may be larger than the maximum of |P(z)| taken over the (2n)th roots of unity, as is shown by the simple example P(z) = z n + ia, a > 0. Following the approach of Frappier et al. [4], Aziz [1] showed that the bound in (1.2) can be considerably improved. In fact, Aziz proved that if P ∈ Pn , then for any real number α, n max P (z) ≤ (Mα + Mα+π ) , |z|=1 2
(1.3)
where Mα = max P ei(α+2lπ )/n , 1≤l≤n
(1.4)
for all real α. In the same paper, Aziz obtained a lower bound for the maximum of |P (z)| on |z| = 1, by proving that if P ∈ Pn , then n 2max P(z) − M0 + Mπ . max P (z) ≥ |z|=1 2 |z|=1
(1.5)
If we restrict ourselves to the class of polynomials having no zeros in |z| < 1, then (1.1) can be replaced by n max P (z) ≤ max P(z), |z|=1 2 |z|=1
(1.6)
whereas if P(z) has no zeros in |z| > 1, then n max P (z) ≥ max P(z). |z|=1 2 |z|=1
(1.7)
Inequality (1.6) was conjectured by Erdös and later proved by Lax [6], whereas inequality (1.7) is due to Turán [14]. Ideally, it is natural to look for improving results in (1.3) when P(z) does not vanish in the unit disk, and accordingly Aziz [1] proved that if P ∈ Pn , and P(z) = 0 in |z| < 1, then for every real number α, 1 n 2 2 2 Mα + Mα+π max P (z) ≤ , |z|=1 2
(1.8)
where Mα is defined by (1.4). As a refinement of (1.8), Rather and Shah [13] proved that if P ∈ Pn , and P(z) = 0 in |z| < 1, then for every real number α, 1
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