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INEQUALITIES FOR COMPLEX POLYNOMIAL WITH RESTRICTED ZEROS
Istayan Das,Robinson Soraisam,Mayanglambam Singhajit Singh,Nirmal Kumar Singha,Barchand Chanam 경남대학교 수학교육과 2023 Nonlinear Functional Analysis and Applications Vol.28 No.4
Let $p(z)$ be a polynomial of degree $n$ and for any complex number $\beta$, let $D_\beta p(z)=np(z)+(\beta-z)p^\prime(z)$ denote the polar derivative of the polynomial with respect to $\beta$. In this paper, we consider the class of polynomial$$p(z)=(z-z_{0})^s\left(a_{0}+\displaystyle{\sum_{\nu=0}^{n-s}a_{\nu}z^{\nu}}\right)$$ of degree $n$ having a zero of order $s$ at $z_0$, $|z_{0}|<1$ and the remaining $n-s$ zeros are outside $|z|<k$, $k\geq1$ and establish upper bound estimates for the maximum of $\left|D_\beta p(z)\right|$ as well as $\left|p(Rz)-p(rz)\right|$, $R\geq r\geq1$ on the unit disk.