Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin

Document Type : Research Paper


Department of Mathematics‎, ‎University of Semnan‎, ‎Semnan‎, ‎Iran.


‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $\alpha$‎, ‎let $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $\alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| \leq k,\ (k\leq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$\alpha\in\mathbb{C}$ with $|\alpha|\geq k$‎,
‎\end{align*} In this paper‎, ‎we obtain a refinement‎ ‎of above inequality‎. ‎Also as an application of our result‎, ‎we extend some inequalities for‎ ‎polar derivative of a polynomial of degree $n$ which‎ ‎does not vanish in $|z|< k$‎, ‎where $k\geq 1$‎, ‎except $s$-fold zeros at the origin‎. 


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