Some Bounds for the Polar Derivative of a Polynomial

الملخص EN

The polar derivative of a polynomial p ( z ) of degree n with respect to a complex number α is a polynomial n p ( z ) + α - z p ′ ( z ) , denoted by D α p ( z ) .

Let 1 ≤ R ≤ k .

For a polynomial p ( z ) of degree n having all its zeros in z ≤ k , we investigate a lower bound of modulus of D α p ( z ) on z = R .

Furthermore, we present an upper bound of modulus of D α p ( z ) on z = R for a polynomial p ( z ) of degree n having no zero in z < k .

In particular, our results in case R = 1 generalize some well-known inequalities.