Some Properties of Rational Functions with Prescribed Poles

http://dx.doi.org/10.4153/CMB-1999-049-0
Canad. Math. Bull. 42(1999), 417-426

  Published:1999-12-01
 Printed: Dec 1999

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.