Vinay Kumar Jain: On the derivative of a self-reciprocal polynomial, 193-196


Similar to a polynomial $P(z)$ with zeros $w_1, w_2, \hdots, w_n$, being called a self-inversive polynomial (see [8]) if

$\displaystyle \{w_1, w_2, \hdots, w_n\} = \{ 1/\overline{w_1},
1/\overline{w_2}, \hdots, 1/\overline{w_n}\},$    

we have called a polynomial $p(z)$ with zeros $z_1, z_2,
\hdots, z_n$, a self-reciprocal polynomial if

$\displaystyle \left\{z_1, z_2, \hdots, z_n\right\} = \left\{ 1/z_1,
1/z_2, \hdots, 1/z_n\right\}$    

and have obtained for a self-reciprocal polynomial $p(z)$ of degree $n$

$\displaystyle \max_{\vert z\vert=1}(\vert p'(z)\vert + \vert p'(\overline{z})\vert) = n
\max_{\vert z\vert=1}\vert p(z)\vert,$    

similar to

$\displaystyle \max_{\vert z\vert=1}\vert P'(z)\vert = \frac{n}{2} \max_{\vert z\vert=1}\vert P(z)\vert,$    

for a self-inversive polynomial $P(z)$ of degree $n$ ([8]).

Key Words: Self-reciprocal polynomial, derivative, unit circle, maximum, self-inversive polynomial.

2010 Mathematics Subject Classification: Primary 30C10; Secondary 30A10.

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