Some Inequalities for the Polar Derivative of Some Classes of Polynomials
In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots (z-z_0)^{t_0}(a_0+\sum\limits_{\nu=\mu} ^{n-(t_m+\cdots+t_0)} a_{\nu}z^\nu)$$ where zeros $z_0,\ldots,z_m$ are in $\{z:|z|<1\}$ and the remaining $n-...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
26.04.2018
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| Subjects | |
| Online Access | Get full text |
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| Summary: | In this paper, we investigate an upper bound of the polar derivative of a
polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots
(z-z_0)^{t_0}(a_0+\sum\limits_{\nu=\mu} ^{n-(t_m+\cdots+t_0)} a_{\nu}z^\nu)$$
where zeros $z_0,\ldots,z_m$ are in $\{z:|z|<1\}$ and the remaining
$n-(t_m+\cdots+t_0 )$ zeros are outside $\{z:|z|<k\}$ where $k \geq 1.$
Furthermore, we give a lower bound of this polynomial where zeros
$z_0,\ldots,z_m$ are outside $\{z:|z|\leq k\}$ and the remaining
$n-(t_m+\cdots+t_0 )$ zeros are in $\{z:|z|<k\}$ where $k\leq 1.$ |
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| DOI: | 10.48550/arxiv.1804.10203 |