International Mathematical Forum, Vol. 8, 2013, no. 11, 501 - 511 HIKARI Ltd, www.m-hikari.com Some Integral Mean Estimates for Polynomials Abdullah Mir, Bilal Ahmad Dar and Q. M. Dawood Department of Mathematics, University of Kashmir Srinagar-190006, India mabdullah mir@yahoo.co.in darbilal67@gmail.com qdawood@gmail.com Copyright c  2013 Abdullah Mir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we establish some integral mean estimates for polynomials P (z)= a n z n + n ∑ υ=μ a n-υ z n-υ , 1 ≤ μ ≤ n, having all its zeros in |z |≤ k, k ≤ 1. Our results not only generalize and refine some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure. Mathematics Subject Classification: 30A10, 30C10, 30C15 Keywords: Polynomial, Maximum modulus, Integral mean estimates 1. INTRODUCTION AND STATEMENT OF RESULTS Let P (z) be a polynomial of degree n and P ′ (z) be its derivative. If P (z) has all its zeros in |z |≤ 1, then it was shown by Turan [9] that Max |z|=1 |P ′ (z)|≥ n 2 Max |z|=1 |P (z)|. (1) Inequality (1) is best possible with equality for P (z)= αz n +β , where |α| = |β |. As an extension of (1) Malik [8] proved that if P (z) has all its zeros in |z |≤ k where k ≤ 1, then Max |z|=1 |P ′ (z)|≥ n 1+ k Max |z|=1 |P (z)|. (2) Aziz [1] obtained a generalization of (2) in the sense that the right hand side of (2) is replaced by a factor involving the integral mean of |P ′ (z)| on |z | = 1.