INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL WITH S-FOLD ZEROS AT THE ORIGIN

‎Let p(z) be a polynomial of degree n and for a complex number α‎, ‎let Dαp(z)=np(z)+(α−z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α‎. ‎Dewan et al proved‎ ‎that if p(z) has all its zeros in |z|≤k, (k≤‎‎1), with s-fold zeros at the origin then for every‎ ‎α∈C with |α|≥k‎,‎‎max|z|=1|Dαp(z)|≥‎‎(n+sk)(|α|−k)1+kmax|z|=1|p(z)|‎.‎In this paper‎, ‎we obtain a refinement‎ ‎of above inequality‎. ‎Also as an application of our result‎, ‎we extend some inequalities for‎ ‎polar derivative of a polynomial of degree n which‎ ‎does not vanish in |z| k‎, ‎where k≥1‎, ‎except s-fold zeros at the origin‎.