Indian J. Pure Appl. Math., 46(3): 337-348, June 2015 c  Indian National Science Academy 10.1007/s13226-015-0133-8 BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS Idrees Qasim and A. Liman Department of Mathematics, National Institute of Technology Srinagar 190 006, India e-mails: idreesf3@gmail.com, abliman@rediffmail.com (Received 27 November 2013; accepted 11 August 2014) In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions. Key words : Rational function; polynomials; inequalities; poles; zeros. 1. I NTRODUCTION Let P n denote the space of complex polynomials of degree at most n and T := {z : |z | =1}. we denote by D - the region inside T and by D + the region outside T. If P ∈ P n , then concerning the estimate of |P 0 (z )| on the unit circle T , we have the following well known result which relates the norm of a polynomial to that of its derivative due to Bernstein [9]. max z∈T |P 0 (z )|≤ n max z∈T |P (z )|. (1.1) The inequality (1.1) is sharp and equality holds for polynomials having all zeros at the origin. The inequality (1.1) was improved by Malik [6]. In fact he proved: If P ∈ P n and Q(z )= z n P ( 1 ¯ z ), then max z∈T |P 0 (z )| + max z∈T |Q 0 (z )|≤ n max z∈T |P (z )|. (1.2) If we consider the class of polynomials P ∈ P n having no zero in D - , then the bounds in inequality (1.1) can be considerably improved. In fact, Erd¨ os conjectured and later Lax [4] verified that if P (z ) does not vanish in D - , then (1.1) can be replaced by max z∈T |P 0 (z )|≤ n 2 max z∈T |P (z )|. (1.3) DOI: