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Self-reciprocal polynomials; integral inequalities.

Let $n\geq 1$ be an integer and let $\mathcal{P}_n$ be the class of polynomials 𝑃 of degree at most 𝑛 satisfying $z^nP(1/z)=P(z)$ for all $z\in C$. Moreover, let 𝑟 be an integer with $1\leq r\leq n$. Then we have for all $P\in\mathcal{P}_n$:

$$\alpha_n(r)\int^{2\pi}_0|P(e^{it})|^2dt\leq\int^{2\pi}_0|P^{(r)}(e^{it})|^2dt\leq\beta_n(r)\int^{2\pi}_0|P(e^{it})|^2dt$$

with the best possible factors

\begin{equation*}\alpha_n(r)=\begin{cases}\prod^{r-1}_{j=0}\left(\frac{n}{2}-j\right)^2, < \text{if 𝑛 is even},\\ \frac{1}{2}\left[\prod^{r-1}_{j=0}\left(\frac{n+1}{2}-j\right)^2+\prod^{r-1}_{j=0}\left(\frac{n-1}{2}-j\right)^2\right], < \text{if 𝑛 is odd},\end{cases}\end{equation*}

and

\begin{equation*}\beta_n(r)=\frac{1}{2}\prod\limits^{r-1}_{j=0}(n-j)^2.\end{equation*}

This refines and extends a result due to Aziz and Zargar (1997).

Horst Alzer¹

1. Morsbacher Str. 10, D-51545 Waldbröl, Germany

Published

April 2010

Manuscript received

14 November 2007