Bernstein-Type Integral Inequalities for a Certain Class of Polynomials-II
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Bernstein-Type Integral Inequalities for a Certain Class of Polynomials-II Abdullah Mir and Abrar Ahmad Abstract. In this paper, we establish some new inequalities in the plane that are inspired by some classical Bernstein-type inequalities that relate the sup-norm of a polynomial to that of its derivative on the unit circle. The obtained results relate the Lγ -norm of the polar derivative and the polynomial. Our results besides derive polar derivative analogues of some classical Bernstein-type inequalities also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials, as well. Mathematics Subject Classification. 30A10, 30C10, 30D15. Keywords. Polar derivative of a polynomial, Bernstein’s inequality, Lγ -norm, Minkowski’s inequality.
1. Introduction
n Let P (z) := v=0 av z v be a polynomial of degree n and P (z) its derivative. The study of Bernstein-type inequalities that relate the norm of a polynomial to that of its derivative and their various versions are a classical topic in analysis. Over a period, these inequalities have been generalized in different domains, in different norms, and for different classes of functions. Here, we study some of the new inequalities centered around Bernstein-type inequalities that relate the Lγ -norm of the polar derivatives and the polynomial under some conditions. For a polynomial P (z) of degree n and α ∈ C, we define: Dα P (z) := nP (z) + (α − z)P (z). Note that Dα P (z) is a polynomial of degree at most n − 1. This is the socalled polar derivative of P (z) with respect to α (see [13]). It generalizes the ordinary derivative in the following sense: Dα P (z) := P (z), lim α→∞ α uniformly with respect to z for |z| ≤ R, R > 0. 0123456789().: V,-vol
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If P (z) is a polynomial of degree n, then concerning the estimate of |P (z)| on the unit disk |z| = 1, we have: max |P (z)| ≤ n max |P (z)|,
|z|=1
(1.1)
|z|=1
whereas concerning the integral-norm estimate of (1.1), we have for every γ≥1: γ1 2π γ γ1 iθ γ 2π P (e ) dθ ≤ n 0 P (eiθ ) dθ . (1.2) 0
Inequality (1.1) is a classical result of Bernstein [14], whereas inequality (1.2) is due to Zygmund [24]. 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). Equality holds in (1.1) and (1.2) only for P (z) = λz n , λ = 0. Noting that these extremal polynomials have all zeros at the origin, so it is natural to seek improvements under appropriate condition on the zeros of P (z). If we restrict ourselves to the class of polynomials having no zeros in |z| < 1, then (1.1) and (1.2) can be improved. In fact, if P (z) is a polynomial of degree n and P (z) = 0 in |z| < 1, then (1.1) and (1.2) can be, respectively, replaced by: n (1.3) max |P (z)| ≤ max |P (z)| 2 |z|=1 |z|=1 and
2π
iθ γ P (e ) dθ
γ1
≤ nCγ
0
2π
P (eiθ )γ dθ
γ1 ,
(1.4)
0
where Cγ =
1 2π
2π
1 + eiβ γ dβ
−1 γ .
0
Inequality (1.3) was conjectur
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