Some lower bounds for the derivative of certain polynomials
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me lower bounds for the derivative of certain polynomials Ishfaq Dar1
· A. Iqbal1
Received: 23 January 2020 / Accepted: 21 August 2020 © Università degli Studi di Ferrara 2020
Abstract For the class of polynomials having no zero outside the disc {z : |z| ≤ k}, there is considerable amount of literature concerning the estimation of lower bound of maximum of |P (z)| in terms of maximum of |P(z)|. However no such results are available when the polynomials have some zeros outside the disc {z : |z| ≤ k}. In this paper we propose to obtain some estimates for the lower bound of maximum modulus of the derivative of a polynomial in terms of maximum modulus of the polynomial when some zeros of the polynomial lie outside the disc {z : |z| ≤ k}. Keywords Polynomials · Lower bounds · Zeros · Inequalities in the complex domain Mathematics Subject Classification 26D10 · 41A17
1 Introduction Polynomials play an important role in many scientific disciplines and computation of the upper and lower bounds of maximum modulus of derivative of a polynomial in terms of maximum modulus of polynomial have important applications in many areas of applied mathematics. The fundamental results concerning the polynomial inequalities can be found in the comprehensive book by Rahman and Schmeisser [8].
B
Ishfaq Dar [email protected] A. Iqbal [email protected]
1
Department of Mathematics, University of Kashmir, Srinagar 190006, India
123
ANNALI DELL’UNIVERSITA’ DI FERRARA
For a positive integer n and 0 < k ≤ 1, we define the following Pn := {P(z) : P(z) =
n
a j z j , an = 0},
j=o
:= {z : |z| ≤ k}, := {z : |z| > 1}. Concerning the estimation of the lower bound of maximum modulus of derivative of a polynomial in terms of maximum modulus of polynomial Turán [9] proved that if P ∈ Pn and P(z) has all zeros in |z| ≤ 1, then max |P (z)| ≥
|z|=1
n max |P(z)|. 2 |z|=1
(1.1)
Equality in inequality (1.1) holds for those polynomials which have all their zeros on |z| = 1. Malik [3] generalized inequality (1.1) and proved that if P ∈ Pn and P(z) has all its zeros in , then max |P (z)| ≥
|z|=1
n max |P(z)|. 1 + k |z|=1
(1.2)
The result is sharp and equality in (1.2) holds for P(z) = (z − k)n . In literature there exist several refinements and generalizations of inequalities (1.1) and (1.2) (see [1,2,5–7]). Recently Rather et al. ([4], Corollary 2.10) improved inequality (1.2), in fact they proved the following sharp result. Theorem A If P ∈ Pn and P(z) has all its zeros in , then max |P (z)| ≥
|z|=1
n max |P(z)| 1 + In |z|=1
(1.3)
where In =
n |an | k 2 + |an−1 | . n |an | + |an−1 |
(1.4)
The result is best possible and equality in (1.4) holds for P(z) = (z − k)n
2 Main results In this paper we propose to relax the condition that all the zeros of the polynomial P ∈ Pn lie in and estimate the lower bound of the maximum of |P (z)| in terms of maximum of |P(z)| for |z| = 1, where P(z) = (z − z 0 )t0 (z − z 1 )t1 ...(z − z k )tk g(z) with z 0 , z 1 , . . . , z k ∈ and remaining n − (t0 + t1 + · · · + tk
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