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1. CMB 1999 (vol 42 pp. 417)
Some Properties of Rational Functions with Prescribed Poles 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 $W^\ast (z) = z^n \overline
{W(1/\bar z)}$. In the paper we consider the case when $k>1$ and
obtain a sharp result. We also show that
$$
\Sup_{|z|=1} \biggl\{ \biggl| \frac{r'(z)}{B'(z)} \biggr| +\biggr|
\frac{\bigl(r^\ast (z)\bigr)'}{B'(z)} \biggr| \biggr\} =
\Sup_{|z|=1} |r(z)|
$$
where $r^\ast (z) = B(z) \overline{r(1/\bar z)}$, and as a consequence of
this result, we present a generalization of a theorem of O'Hara and
Rodriguez for self-inversive polynomials. Finally, we establish a similar
result when supremum is replaced by infimum for a rational function which
has all its zeros in the unit circle.
Category:26D07 |