ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2016, Vol. 37, No. 2, pp. 155–159. c  Pleiades Publishing, Ltd., 2016. On an Inequality of Paul Turan Concerning Polynomials Abdullah Mir 1* , K. K. Dewan 2** , and Imtiaz Hussain 1*** (Submitted by S. N. Tronin) 1 Department of Mathematics, University of Kashmir, Srinagar-19006, India 2 Department of Mathematics, Faculty of Natural Sciences Jamia Millia Islamia (Central University), New Delhi-110025 Received December 24, 2014 Abstract—Let P (z ) be a polynomial of degree n and P s (z ) be its sth derivative. In this paper, we shall prove some inequalities for the sth derivative of a polynomial having zeros inside a circle, which as a special case give generalizations and refinements of some results of Turan, Govil, Malik and others. DOI: 10.1134/S1995080216020104 Keywords and phrases: Inequality, sth derivative of a polynomial. 1. INTRODUCTION Let P (z) be a polynomial of degree n and P ′ (z) its derivative. It was shown by Turan [8] that if P (z) has all its zeros in |z|≤ 1, then max |z|=1 |P ′ (z)|≥ n 2 max |z|=1 |P (z)|. (1.1) Inequality (1.1) was refined by Aziz and Dawood [3], who under the same hypothesis proved that max |z|=1 |P ′ (z)|≥ n 2  max |z|=1 |P (z)| + min |z|=1 |P (z)|  . (1.2) Both the inequalities (1.1) and (1.2) are sharp and equality holds for P (z)= αz n + β, where |α| = |β|. In the literature, there exists a variety of generalizations and refinements of inequalities (1.1) and (1.2) (for example see Malik [7], Mir, Dar and Baba [1], Jain [6] etc). As an extension of (1.1), Malik [7] showed that if P (z) has all its zeros |z|≤ k, k ≤ 1, then max |z|=1 |P ′ (z)|≥ n 1+ k max |z|=1 |P (z)|, (1.3) whereas for P (z) having all its zeros in |z|≤ k,k ≤ 1 with s-fold zeros at the origin, Aziz and Shah [4] proved that max |z|=1 |P ′ (z)|≥ n + sk 1+ k max |z|=1 |P (z)|. (1.4) Both the estimates (1.3) and (1.4) are also sharp and equality in (1.3) holds for P (z)=(z + k) n whereas equality in (1.4) holds for P (z)= z s (z + k) n−s , 0 ≤ s ≤ n. * E-mail: mabdullah_mir@yahoo.co.in ** E-mail: kkdewan123@yahoo.co.in *** E-mail: imtiaz5096@yahoo.com 155