Integral Inequalities for Self-Reciprocal Polynomials

Proceedings Mathematical Sciences

Horst Alzer¹ 

[1] Morsbacher Str. 0, D- Waldbröl, Germany$$

关键词: Self-reciprocal polynomials;    integral inequalities.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let $n≥ 1$ 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 $zin C$. Moreover, let 𝑟 be an integer with $1≤ r≤ n$. Then we have for all $Pinmathcal{P}_n$:$$𝛼_n(r)int^{2𝜋}_0|P(e^{it})|^2dt≤int^{2𝜋}_0|P^{(r)}(e^{it})|^2dt≤𝛽_n(r)int^{2𝜋}_0|P(e^{it})|^2dt$$with the best possible factorsegin{equation*}𝛼_n(r)=egin{cases}prod^{r-1}_{j=0}left(frac{n}{2}-jight)^2, < ext{if 𝑛 is even}, frac{1}{2}left[prod^{r-1}_{j=0}left(frac{n+1}{2}-jight)^2+prod^{r-1}_{j=0}left(frac{n-1}{2}-jight)^2ight], < ext{if 𝑛 is odd},end{cases}end{equation*}andegin{equation*}𝛽_n(r)=frac{1}{2}prodlimits^{r-1}_{j=0}(n-j)^2.end{equation*}This refines and extends a result due to Aziz and Zargar (1997).

【 授权许可】

Unknown