Some Upper Bound Estimates for the Maximal Modulus of the Polar Derivative of a Polynomial
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AND COMPLEX ANALYSIS
Some Upper Bound Estimates for the Maximal Modulus of the Polar Derivative of a Polynomial A. Mir1* and M. Ibrahim Sheikh2** 1
2
University of Kashmir, Srinagar, India Pusan National University, Busan, Republic of Korea
Received September 22, 2018; revised February 12, 2019; accepted April 25, 2019
Abstract—This paper deals with the problem of finding some upper bound estimates for the maximal modulus of the polar derivative of a complex polynomial on a disk under certain constraints on the zeros and on the functions involved. A variety of interesting results follow as special cases from our results. MSC2010 numbers : 30A10, 30C10, 30C15 DOI: 10.3103/S1068362320030085 Keywords: complex polynomial; polar derivative; maximum modulus; zeros.
1. INTRODUCTION Let Pn denote the space of all complex polynomials P (z) :=
n
aj z j of degree n and P (z) is the
j=0
derivative of P (z). A famous result known as Bernstein’s inequality (for reference, see [3]) states that if P ∈ Pn , then maxP (z) ≤ nmaxP (z), (1.1) |z|=1
|z|=1
where as concerning the maximum modulus of P (z) on the circle |z| = R ≥ 1, we have (for reference see [11]), (1.2) max P (z) ≤ Rn maxP (z). |z|=R
|z|=1
Both the above inequalities are sharp and equality in each holds only whenP (z) is a constant multiple of zn. It was observed by Bernstein [3] that (1.1) can be deduced from (1.2), by making use of Gauss–Lucas theorem and the proof of this fact was given by Govil et al. [4]. If we restrict ourselves to the class of polynomials P ∈ Pn , with P (z) = 0 in |z| < 1, then (1.1) and (1.2) can be respectively replaced n (1.3) maxP (z) ≤ maxP (z), 2 |z|=1 |z|=1 and
Rn + 1 maxP (z). max P (z) ≤ 2 |z|=1 |z|=R≥1
(1.4)
¨ and later proved by Lax [8], where as inequality (1.4) was Inequality (1.3) was conjectured by Erdos proved by Ankeny and Rivlin [1], for which they made use of (1.3). * **
E-mail: [email protected] E-mail: [email protected]
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MIR, IBRAHIM SHEIKH
Inequality (1.1) can be seen as a special case of the following inequality which is also due to Bernstein [3]. Theorem A. Let F ∈ Pn , having all its zeros in |z| ≤ 1 and f (z) be a polynomial of degree at most n. If |f (z)| ≤ |F (z)| for |z| = 1, then for |z| ≥ 1, we have f (z) ≤ F (z). (1.5) Equality holds in (1.5) for f (z) = eiη F (z), η ∈ R. Inequality (1.1) can be obtained from inequality (1.5) by taking F (z) = M z n , where M = max|f (z)|. |z|=1
In the same way, inequality (1.2) follows from a result which is a special case of Bernstein–Walsh lemma ([10], Corollary 12.1.3). Theorem B. Let F ∈ Pn , having all its zeros in |z| ≤ 1 and f (z) be a polynomial of degree at most n. If |f (z)| ≤ |F (z)| for |z| = 1, then f (z) < F (z), for |z| > 1, unless f (z) = eiη F (z) for some η ∈ R. In 2011, Govil et al. [5] proved a more general result which provides a compact generalization of inequalities (1.1), (1.2), (1.3), and (1.4) and includes Theorem A and Theore
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