Inequalities for the Polar Derivative of a Polynomial

Abstract EN

For a polynomial p(z) of degree n, we consider an operator Dα which map a polynomial p(z) into Dαp(z):=(α-z)p'(z)+np(z) with respect to α.

It was proved by Liman et al.

(2010) that if p(z) has no zeros in |z|<1, then for all α,  β∈C with |α|≥1,   |β|≤1 and |z|=1, |zDαp(z)+nβ((|α|-1)/2)p(z)|≤(n/2){[|α+β((|α|-1)/2)|+|z+β((|α|-1)/2)|]max|z|=1|p(z)|-[|α+β((|α|-1)/2)|-|z+β((|α|-1)/2)|]min|z|=1|p(z)|}.

In this paper we extend the above inequality for the polynomials having no zeros in |z|

Our result generalizes certain well-known polynomial inequalities.