• Integral Inequalities for Self-Reciprocal Polynomials

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      https://www.ias.ac.in/article/fulltext/pmsc/120/02/0131-0137

    • Keywords

       

      Self-reciprocal polynomials; integral inequalities.

    • Abstract

       

      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).

    • Author Affiliations

       

      Horst Alzer1

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

       
  • Proceedings – Mathematical Sciences | News

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