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

Similar to a polynomial

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

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

of degree

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

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

of degree

([8]).