Inequalities involving polar derivative and N-operator

Abstract

In this paper, we obtain some inequalities involving linear differential operator N for polynomials over complex domain and study the dependence of modulus of \(P(Rz)-\beta P(rz)+\delta \{ (\frac{R+1}{r+1})^n-|\beta |\}P(rz)\) for all complex numbers \(\alpha , \beta \) and \( \delta \) with \(|\alpha |\ge 1, |\beta |\le 1, |\delta |\le 1\) on the extreme values of |P(z)| over the boundary of unit circle after successive applications of \(D_{\alpha }\) and N. Our results besides yielding some interesting inequalities as special cases also generalize recently obtained inequalities related to the content.