NEW ZEALAND JOURNAL OF MATHEMATICS Volume 44 (2014), 83-91 SOME INTEGRAL MEAN ESTIMATES FOR POLYNOMIALS N. A. Rather, Suhail Gulzar, and K. A. Thakur (Received 2 May, 2014) Abstract. For the class of Lacunary polynomials P (z)= anz n + ∑ n ν=μ a n-ν z n-ν , 1 ≤ μ ≤ n, of degree n having all their zeros in |z|≤ k where k ≤ 1, Aziz and Shah [6] proved for each r> 0 n ( 2π Z 0   P  e iθ    r dθ ) 1 r ≤ ( 2π Z 0   1+ k μ e iθ    r dθ ) 1 r max |z|=1 |P 0 (z)|. In this paper, we extend above inequality to the polar derivative thereby estab- lish some refinements and generalizations of some known polynomial inequal- ities concerning the polar derivative of a polynomial with restricted zeros. 1. Introduction and Statement of Results Let P (z) be a polynomial of degree n. It was shown by Tur´ an [13] that if P (z) has all its zeros in |z|≤ 1, then n max |z|=1 |P (z)|≤ 2 max |z|=1 |P 0 (z)| . (1) Inequality (1) is best possible with equality holds for P (z)= αz n + β where |α| = |β|6 =0. As an extension of (1), Malik [9] proved that if P (z) is a polynomial of degree n having all its zeros in |z|≤ k where k ≤ 1, then n max |z|=1 |P (z)|≤ (1 + k) max |z|=1 |P 0 (z)| . (2) Equality in (2) holds for P (z)=(z + k) n where k ≤ 1. On the other hand, for the class of polynomials P (z)= a n z n + ∑ n j=μ a n-j z n-j , 1 ≤ μ ≤ n, of degree n having all their zeros in |z|≤ k, k ≤ 1, Aziz and Shah [6] proved that max |z|=1 |P 0 (z)|≥ n 1+ k μ  max |z|=1 |P (z)| + 1 k n-μ min |z|=k |P (z)|  . (3) Malik [10] obtained a generalization of (1) in the sense that the left-hand side of (1) is replaced by a factor involving the integral mean of |P (z)| on |z| =1. In fact, he proved that if P (z) is a polynomial of degree n having all its zeros in |z|≤ 1, then for each r> 0, n ( 2π Z 0   P ( e iθ )  r dθ ) 1 r ≤ ( 2π Z 0   1+ e iθ   r dθ ) 1 r max |z|=1 |P 0 (z)|. (4) 2010 Mathematics Subject Classification 30C10, 30A10, 41A17. Key words and phrases: Polynomials; Inequalities in the complex domain; Polar derivative.