SOME INTEGRAL INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL

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,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-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.