On the Inequalities Concerning Polynomials
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Complex Analysis and Operator Theory
On the Inequalities Concerning Polynomials Prasanna Kumar1 Received: 22 February 2020 / Accepted: 12 August 2020 © Springer Nature Switzerland AG 2020
Abstract If P(z) = an nj=1 (z − z j ) is a complex polynomial of degree n having all its zeros in |z| ≤ K , K ≥ 1 then Aziz (Proc Am Math Soc 89:259–266, 1983) proved that max |P (z)| ≥
|z|=1
n K 2 max |P(z)|. 1 + Kn K + |z j | |z|=1
(0.1)
j=1
In this paper we sharpen the inequality (0.1) and further extend the obtained result to the polar derivative of a polynomial. As a consequence we also derive two results on the generalization of Erdös–Lax type inequality for the class of polynomials having no zeros in the disc |z| < K , K ≤ 1. Keywords Inequalities · Polynomials · Zeros Mathematics Subject Classification 30A10
1 Introduction and Statement of Results If P(z) is a polynomial of degree n then from a well-known inequality due to Bernstein [3], we have max |P (z)| ≤ n max |P(z)|.
|z|=1
|z|=1
Communicated by Dan Volok.
B 1
Prasanna Kumar [email protected] Department of Mathematics, Birla Institute of Technology and Science Pilani K K Birla Goa Campus, Sancoale, Goa 403726, India 0123456789().: V,-vol
(1.1)
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P. Kumar
The inequality (1.1) is sharp and equality holds, if P(z) has all its zeros at the origin. If P(z) is a polynomial of degree n having no zeros in |z| < 1, then Erdös conjectured and later Lax [12] proved that max |P (z)| ≤
|z|=1
n max |P(z)|. 2 |z|=1
(1.2)
The inequality (1.2) is best possible and equality holds for P(z) = a + bz n , where |a| = |b|. If P(z) is a polynomial of degree n having all its zeros in |z| ≤ 1, then Turán [17] proved that max |P (z)| ≥
|z|=1
n max |P(z)|. 2 |z|=1
(1.3)
Aziz [1] considered the modulus of each zero of the underlying polynomial in the bound and generalized the inequality (1.3) to the class of polynomials having all their zeros in a closed discof finite radius greater than or equal to unit length by proving that, if P(z) = an nj=1 (z − z j ) is a complex polynomial of degree n with |z j | ≤ K , K ≥ 1, then max |P (z)| ≥
|z|=1
n K 2 max |P(z)|. n 1+K K + |z j | |z|=1
(1.4)
j=1
Recently Govil and Kumar [8] generalized the inequality (1.3) to the class of polynomials having all their zeros in the disc |z| ≤ K , K ≥ 1, by including the information from leading and constant coefficients of the polynomial, but it did not capture the modulus of each individual zero, which plays a crucial role in sharpening the bound. We prove a generalization of (1.3) to the class of polynomials having all their zeros in the disc |z| ≤ K , K ≥ 1, by obtaining the bound which involves the modulus of each zero of the underlying polynomial, and at the same time our result sharpens (1.4) and also several of the earlier results considerably. Theorem 1.1 If P(z) = a0 + a1 z + · · · + an−1 z n−1 + an z n = an nj=1 (z − z j ) is a polynomial of degree n which has all its zeros in the disk |z| ≤ K , K ≥ 1, then max |P (z)| ≥
|z|=1
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