CREAT. MATH. INFORM. Volume 25 (2016), No. 2, Pages 197 - 203 Online version at https://creative-mathematics.cunbm.utcluj.ro/ Print Edition: ISSN 1584 - 286X; Online Edition: ISSN 1843 - 441X DOI: https://doi.org/10.37193/CMI.2016.02.11 Rate of growth of polynomials with restricted zeros ABDULLAH MIR and Q. M. DAWOOD ABSTRACT. In this paper we consider for a fixed μ, the class of polynomials P (z)= a 0 + n ∑ ν=μ aν zν , 1 ≤ μ ≤ n, of degree at most n not vanishing in the disk |z| < k,k > 0. For any ρ>σ ≥ 1 and 0 <r ≤ R ≤ k, we investigate the dependence of ‖ P (ρz) − P (σz) ‖ R on ‖ P ‖r and derive various refinements and generalizations of some well known results. 1. I NTRODUCTION Let P n be the class of polynomials P (z)= n ∑ ν=0 a ν z ν of degree at most n. For P ∈ P n , we define ‖ P ‖:= max |z|=1 |P (z)|, ‖ P ‖ R := max |z|=R |P (z)|, ‖ P (ρz) − P (σz) ‖ R := max |z|=R |P (ρz) − P (σz)| and m := min |z|=k |P (z)|. If P ∈ P n , then concerning the estimate of the maximum of |P ′ (z)| on the unit circle |z| =1 and the estimate of the maximum of |P (z)| on a larger circle |z| = R> 1, we have ‖ P ′ ‖≤ n ‖ P ‖ (1.1) and ‖ P ‖ R ≤ R n ‖ P ‖ . (1.2) Inequality (1.1) is a well-known result of S. Bernstein (for reference see [15, p-508]), whereas inequality (1.2) is a simple deduction from maximum modulus principle (see [15, p-405]). If we restrict ourselves to the class of polynomials P ∈ P n with P (z) =0 in |z| < 1, then Erd ¨ os conjectured and later Lax (for reference see [15, p-562]), verified that the inequality (1.1) can be replaced by ‖ P ′ ‖≤ n 2 ‖ P ‖ . (1.3) As an extension of (1.3), it was shown by Malik (for reference see [15, p-563]), that if P ∈ P n and P (z) =0 in |z| <k,k ≥ 1, then ‖ P ′ ‖≤ n 1+ k ‖ P ‖ . (1.4) Received: 14.01.2016. In revised form: 30.03.2016. Accepted: 08.04.2016 2010 Mathematics Subject Classification. 30A10, 30C10, 30C15. Key words and phrases. Polynomials, inequalities, maximum modulus principle, Growth. Corresponding author: Abdullah Mir; mabdullah mir@yahoo.co.in 197