Some $L^{γ}$ Inequalities for the Polar Derivative of a Polynomial (Analysis in Theory and Applications, 2017)
In this paper, we consider an operator $D_α$ which maps a polynomial $P(z)$ in to $D_αP(z):= np(z)+(α−z)P′(z)$, where $α ∈ \mathcal{C}$ and obtain some $L^{γ}$ inequalities forlucanary polynomials having zeros in $|z|≤k≤1$. Our results yields several generalizationsand refinements of many known results and also provide an alternative proof ofa result due to Dewan et al. [7], which is independent of Laguerre’s theorem.
Keywords: Polar derivative; polynomials; $L^{γ}$-inequalities in the complex domain; Laguerre's theorem.