Integral inequality for the polar derivatives of polynomials

Document Type : Research Paper


School of Mathematics, Renmin University of China, Beijing, 100872, China


Let $P(z)$ be a polynomial of degree $n$ and for any complex number $\alpha$, let $$D_{\alpha}P(z)= nP(z) + (\alpha - z)P^{\prime}(z)$$ denote the polar derivative of $P(z)$ with respect to a complex number $\alpha$.   In this paper, we prove some $L_{r}$ inequalities for the polar derivative of a polynomial have all zeros in $|z| \leq  1$. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, {\it Inequalities for the polar derivative of a polynomial}, J. Interd. Math.  10 (2007), no. 4,  525--531] and includes as special cases several interesting many known results.