Indian J. Pure Appl. Math., 51(2): 749-760, June 2020 c  Indian National Science Academy DOI: 10.1007/s13226-020-0428-2 CERTAIN ESTIMATES OF THE DERIVATIVE OF A MEROMORPHIC FUNCTION ON BOUNDARY OF THE UNIT DISK Abdullah Mir Department of Mathematics, University of Kashmir, Srinagar 190 006, India e-mail: drabmir@yahoo.com (Received 12 February 2019; accepted 12 April 2019) In this paper we establish a family of comparison inequalities between rational functions with prescribed poles for the sup-norms on the unit circle in the complex plane. Certain estimates for the modulus of the derivative of rational functions as well as some inequalities of Tur´ an’s type are also obtained. The obtained results produce many inequalities for polynomials and polar derivatives as special cases. Key words : Rational functions; Bernstein inequality; poles; zeros. 2010 Mathematics Subject Classification : 30A10, 30C10, 26D10. 1. I NTRODUCTION Let P n denote the class of all complex polynomials of degree n. A classical majorization result due to Bernstein is that, for two polynomials P and Q with Q ∈ P n , deg P ≤ deg Q and Q(z ) =0 for |z | > 1, the majorization |P (z )|≤|Q(z )| on the unit circle |z | =1 implies the majorization of their derivatives |P ′ (z )|≤|Q ′ (z )|. In particular, this majorization result allows to establish famous Bernstein inequality [3] for the sup-norms on the unit circle: for P ∈ P n , it is true that max |z|=1 |P ′ (z )|≤ n max |z|=1 |P (z )|. (1.1) The above inequality (1.1) was proved by Bernstein in 1912. Later in 1985, Malik and Vong [7] used a parameter β and proved the following Bernstein-type inequality from which (1.1) can also be deduced.