Some inequalities for polynomials with restricted zeros
- PDF / 256,512 Bytes
- 7 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 206 Views
me inequalities for polynomials with restricted zeros N. A. Rather1 · Ishfaq Dar1
· A. Iqbal1
Received: 15 August 2020 / Accepted: 23 October 2020 © Università degli Studi di Ferrara 2020
Abstract By using the boundary Schwarz lemma, it was shown by Dubinin (J Math Sci 143:3069–3076, 2007) that if P(z) is a polynomial of degree n having all its zeros in |z| ≤ 1, then for all z on |z| = 1 for which P(z) = 0,
z P (z) Re P(z)
n 1 ≥ + 2 2
|an | − |a0 | . |an | + |a0 |
In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we also obtain sharp refinements of some known results due to Malik (J Lond Math Soc 1:57– 60, 1969), Aziz and Rather (Math Ineq Appl 1:231–238, 1998). These results take into account the size of the constant term and the leading coefficient of the polynomial P(z). Keywords Polynomials · Inequalities · Refinement Mathematics Subject Classification 26D10 · 41A17 · 30C15
1 Introduction and main results In scientific investigations the experimental observations when translated into mathematical language lead to mathematical models. The solution of these models could require estimating how large or small the maximum modulus of the derivative of an
B
Ishfaq Dar [email protected] N. A. Rather [email protected] A. Iqbal [email protected]
1
Department of Mathematics, University of Kashmir, Srinagar 190006, India
123
ANNALI DELL’UNIVERSITA’ DI FERRARA
algebraic polynomial can be in terms of the maximum modulus of that polynomial. Bounds for such type of problems are of some practical importance. Since there are no closed formulae for precise evaluation of these bounds and whatever is available in the literature is in the form of approximations. These approximate bounds, when computed efficiently, are quite satisfactory for the needs of investigators and scientists. Therefore there is always a desire to look for better and improved bounds than those available in the literature. It is this aspiration of obtaining more refined and revamped bounds that have inspired our work in this article. In this paper we have generalized and refined some well-known results concerning the polynomials due to Turán [11], Dubinin [5], and others. To begin with let Pn denote the class of all polynomials n over the P(z) = an z n + an−1 z n−1 + · · · + a0 of degree complex field C. Concerning the estimation of the lower bound of Re z P (z)/P(z) on |z| = 1, Dubinin [5] proved the following result. Theorem A If P ∈ Pn having all its zeros in |z| ≤ 1, then for all z on |z| = 1 for which P(z) = 0 Re
z P (z) P(z)
≥
n 1 |an | − |a0 | 1+ . 2 n |an | + |a0 |
(1.1)
It is worth mentioning here that the proof of the above result is based on boundary Schwarz lemma due to R. Osserman [9]. Here we propose to generalize the above result by using simple techniques, as a consequence of which we present a simple and direct proof of Theorem A, without invoking the boundary Schwarz lem
Data Loading...
