EXTENSION OF TURÁN-TYPE INEQUALITIES FOR POLAR DERIVATIVES OF POLYNOMIALS INTO INTEGRAL MEAN VERSION

Let p ( z ) be a polynomial of degree n and let D (alpha) p ( z ) = np ( z ) + (alpha- z ) p ' ( z ) denote the polar derivative of the polynomial p ( z ) with respect to a real or complex number alpha . If p ( z ) is a polynomial of degree n having all its zeros in z 6k, k 1, then for a real or complex number alpha with alpha k , Aziz and Rather [J. Math. Ineq. Appl. 1 (1998), 231-238] proved max( |z|=1 )| D (alpha) p ( z )| <downwards arrow with double stroke> n ( |alpha| - k / 1+ k(n ) ) max(|z| =1) | p(z) | We first extend the above inequality into integral mean without applying subordination property. As an application of our result, we prove another integral mean inequality. Our results have interesting consequences to the earlier wellknown inequalities.