Abstract view
Some Properties of Rational Functions with Prescribed Poles


Published:19991201
Printed: Dec 1999
Abdul AzizUlAuzeem
B. A. Zarger
Abstract
Let $P(z)$ be a polynomial of degree not exceeding $n$ and let
$W(z) = \prod^n_{j=1}(za_j)$ 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 selfinversive 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.