Inequalities for the polar derivative of a polynomial
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ORIGINAL RESEARCH PAPER
Inequalities for the polar derivative of a polynomial M. H. Gulzar1 • B. A. Zargar1 • Rubia Akhter1 Received: 30 May 2019 / Accepted: 3 January 2020 Ó Forum D’Analystes, Chennai 2020
Abstract Let P(z) be a polynomial of degree n having all its zeros in jzj 1, then according to Turan (Compositio Mathematica 7:89–95, 2004) n max jP0 ðzÞj max jPðzÞj: 2 jZj¼1 jZj¼1 In this paper, we shall use polar derivative and establish a generalisation and an extension of this result. Our results also generalize variety of other results. Keywords Polynomial Polar derivative Inequalities
Mathematics Subject Classification 30A10 30C15
1 Introduction Let P n denote the class of all complex polynomials of degree at most n. Let B ¼ fz; jzj ¼ 1g denotes the unit disk and B and Bþ denote the regions inside and outside the disk B respectively. If P 2 P n , then according to the well known result of Bernstein [4] max jP0 ðzÞj n max jPðzÞj: z2B
z2B
ð1Þ
Inequality (1) is best possible and equality holds for the polynomial PðzÞ ¼ kzn ;
& M. H. Gulzar [email protected] B. A. Zargar [email protected] Rubia Akhter [email protected] 1
Department of Mathematics, Kashmir University, Srinagar 190006, India
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where k is a complex number. If we restrict ourselves to the class of polynomials having no zeros in B [ B , then it was conjectured by Erdo¨s and later on proved by Lax [6] that max jP0 ðzÞj z2B
n max jPðzÞj; 2 z2B
ð2Þ
and if P has no zero in B [ Bþ ; then it was proved by Turan [8] that max jP0 ðzÞj z2B
n max jPðzÞj: 2 z2B
ð3Þ
The inequalities (2) and (3) are also best possible and equality holds for polynomials which have all zeros on B. If P(z) is a polynomial of degree n and a a complex number, then the polar derivative of P(z) with respect to a, denoted by Da PðzÞ is defined by Da PðzÞ ¼ nPðzÞ þ ða zÞP0 ðzÞ: Clearly Da PðzÞ is a polynomial of degree at most n 1 and it generalizes the ordinary derivative in the sense that lim
a!1
Da PðzÞ ¼ P0 ðzÞ: a
As an extension of (1), Aziz and Shah [3] used polar derivative and established that if P(z) is a polynomial of degree n, then for every real or complex number a with jaj [ 1 and for z 2 B, jDa PðzÞj njaj max jPðzÞj: z2B
ð4Þ
Aziz [1] extended inequality (2) to the polar derivative and proved that if p is a polynomial of degree n having all zero in z 2 B [ Bþ then for a 2 C with jaj 1 max jDa PðzÞj z2B
nðjaj þ 1Þ max jPðzÞj: z2B 2
ð5Þ
If we divide the two sides of (4) and (5) by jaj and let jaj ! 1, we get inequalities (1) and (2) respectively. Shah [7] extended (3) to the polar derivative and proved the following result: Theorem 1.1 If P 2 P n and has all zeros in z 2 B [ B , then for jaj 1 max jDa PðzÞj z2B
nðjaj 1Þ max jPðzÞj: z2B 2
ð6Þ
Theorem (1.1) generalizes (3) and to obtain (3), divide both sides of Theorem (1.1) by jaj and let jaj ! 1.
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2 Main results In this paper we
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