Inequalities for the polar derivative of a polynomial with S-fold zeros at the origin

ýLet p(z) be a polynomial of degree n and for a complex number αý, ýlet Dαp(z)=np(z)(α−z)p′(z) denote the polar derivative of the polynomial p(z) with respect to αý. ýDewan et al provedý ýthat if p(z) has all its zeros in |z|≤k, (k≤ýý1), with s-fold zeros at the origin then for everyý ýα∈C with |α|≥ký,
ýýmax|z|=1|Dαp(z)|≥ýý(n)(|α|−k)1鉦|z|=1|p(z)|ý.ý
In this paperý, ýwe obtain a refinementý ýof above inequalityý. ýAlso as an application of our resultý, ýwe extend some inequalities forý ýpolar derivative of a polynomial of degree n whichý ýdoes not vanish in |z|