A Zygmund-type integral inequality for polynomials
- PDF / 287,707 Bytes
- 9 Pages / 595.276 x 790.866 pts Page_size
- 34 Downloads / 186 Views
Arabian Journal of Mathematics
Abdullah Mir
A Zygmund-type integral inequality for polynomials
Received: 28 October 2018 / Accepted: 5 March 2019 © The Author(s) 2019
Abstract Let P(z) be a polynomial of degree n which does not vanish in |z| < 1. Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that s (s) z P + β n s P(z) ≤ n s 1 + β + β max|P(z)|, s s 2 2 2 2s |z|=1 for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. The L γ analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved 0
2π
γ γ1 ns isθ (s) iθ iθ e P (e ) + β s P(e ) dθ 2
≤ ns 0
2π
2π P(eiθ )γ dθ
γ1
γ1 0 β γ β iα 1 + s e + s dα γ1 , 2 2 γ
2π 1 + eiα dα 0
where n s = n(n − 1) . . . (n − s + 1) and 0 ≤ γ < ∞. In this paper, we generalize this and some other related results. Mathematics Subject Classification
30A10 · 30C10 · 30C15
1 Introduction Let Pn be the class of polynomials P(z) = P ∈ Pn , we have
n
v=0 av z
v
of degree n and P (s) (z) be its sth derivative. For
max |P (z)| ≤ n max |P(z)|
|z|=1
|z|=1
(1.1)
and for every γ ≥ 1, A. Mir (B) Department of Mathematics, University of Kashmir, Srinagar 190006, India E-mail: [email protected]
123
Arab. J. Math.
2π
0
γ1 iθ γ (e ) dθ ≤ n P
2π
0
γ γ1 iθ P(e ) dθ .
(1.2)
The inequality (1.1) is a classical result of Bernstein [10], whereas the inequality (1.2) is due to Zygmund [13] who proved it for all trigonometric polynomials of degree n and not only for those of the form P(eiθ ). Arestov [1] proved that (1.2) remains true for 0 < γ < 1 as well. If we let γ → ∞ in (1.2), we get (1.1). The above two inequalities (1.1) and (1.2) can be sharpened, if we restrict ourselves to the class of polynomials having no zeros in |z| < 1. In fact, if P ∈ Pn and P(z) = 0 in |z| < 1, then (1.1) and (1.2) can be, respectively, replaced by max |P (z)| ≤
|z|=1
n max |P(z)| 2 |z|=1
(1.3)
and 0
2π
γ1 iθ γ P (e ) dθ ≤ nC γ
0
2π
γ γ1 iθ P(e ) dθ ,
(1.4)
where Cγ =
1 2π
0
2π
γ −1 γ iα . 1 + e dα
(1.5)
The inequality (1.3) was conjectured by Erdös and later proved by Lax [9], whereas (1.4) was proved by De-Bruijn [4] for γ ≥ 1. Further, Rahman and Schmeisser [11] have shown that (1.4) holds for 0 < γ < 1 as well. If we let γ → ∞ in inequality (1.4), we get (1.3). As an extension of (1.3), Jain [7] proved that if P ∈ Pn and P(z) = 0 in |z| < 1, then n nβ β β (1.6) P(z) ≤ 1 + + max|P(z)|, z P (z) + 2 2 2 2 |z|=1 for |z| = 1 and for every β ∈ C with |β| ≤ 1. In 2000, Jain [8] further improved (1.6) by obtaining under the same hypothesis that n nβ β β P(z) ≤ 1 + + max|P(z)| z P (z) + 2 2 2 2 |z|=1 β β − 1 + − min |P(z)| , 2 2 |z|=1
(1.7)
for |z| = 1 and for every β ∈ C with |β| ≤ 1. Recently, Hans and Lal [
Data Loading...
