Some Properties of Rational Functions with Prescribed Poles

Canadian mathematical bulletin

关键词: asymptotic bound;    circle packing;   

DOI  :  10.4153/CMB-1999-049-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $P(z)$ be a polynomial of degree not exceeding $n$ and let$W(z) = prod^n_{j=1}(z-a_j)$ where $|a_j|> 1$, $j =1,2,dots, n$. If the rational function $r(z) = P(z)/W(z)$ does not vanish in $|z|< k$, then for $k=1$ it is known that$$|r'(z)| leq frac{1}{2} |B'(z)| Sup_{|z|=1} |r(z)|$$where $B(Z) = W^ast(z)/W(z)$ and $W^ast (z) = z^n overline{W(1/ar z)}$. In the paper we consider the case when $k>1$ andobtain a sharp result. We also show that$$Sup_{|z|=1} iggl{ iggl| frac{r'(z)}{B'(z)} iggr| +iggr|frac{igl(r^ast (z)igr)'}{B'(z)} iggr| iggr} =Sup_{|z|=1} |r(z)|$$where $r^ast (z) = B(z) overline{r(1/ar z)}$, and as a consequence ofthis result, we present a generalization of a theorem of O'Hara andRodriguez for self-inversive polynomials. Finally, we establish a similarresult when supremum is replaced by infimum for a rational function whichhas all its zeros in the unit circle.

【 授权许可】

Unknown