Anal. Theory Appl. Vol. 27, No. 4 (2011), 340–350 DOI10.1007/s10496-011-0340-z SOME INTEGRAL INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL Abdullah Mir (Kashmir University, India) Sajad Amin Baba (Govt. Hr. Sec. Institute, India) Received Oct. 9, 2010 c  Editorial Board of Analysis in Theory & Applications and Springer-Verlag Berlin Heidelberg 2011 Abstract. If P(z) is a polynomial of degree n which does not vanish in |z| < 1, then it is recently proved by Rather [Jour. Ineq. Pure and Appl. Math., 9 (2008), Issue 4, Art. 103] that for every γ > 0 and every real or complex number α with |α |≥ 1,   2π 0 |D α P(e iθ )| γ dθ  1/γ ≤ n(|α | + 1) C γ   2π 0 |P(e iθ )| γ dθ  1/γ , C γ =  1 2π  2π 0 |1 + e iβ | γ dβ  -1/γ , where D α P(z) denotes the polar derivative of P(z) with respect to α . In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 (1988), 306-313] as a special case. Key words: polar derivative, polynomial, Zygmund inequality, zeros AMS (2010) subject classification: 30A10, 30C10, 30D15, 41A17 1 Introduction and Statement of Results Let P(z)= n ∑ v=0 a v z v be a polynomial of degree atmost n and P ′ (z) its derivative, then max |z|=1 |P ′ (z)|≤ n max |z|=1 |P(z)|, (1.1) and for every γ ≥ 1,   2π 0 |P ′ (e iθ )| γ |rmd θ  1/γ ≤ n   2π 0 |P(e iθ )| γ dθ  1/γ . (1.2)