CERTAIN NEW RESULTS ON RATIONAL FUNCTIONS WITH PRESCRIBED POLES

Abstract

Let $R_{n}$ be the space of rational functions with prescribed poles. If $r \in R_{n},$ does not vanish in $|z|<k,$ then for $k=1$ $$|r^{\prime}(z)|\le \frac{|B^{\prime}(z)|}{2}\sup_{z\in T}|r(z)|,$$ where $B(z)$ is the Blaschke product.

In this paper, we consider a more general class of rational functions $rof \in R_{m^{\star}n}, $ defined by $$(rof)(z)=r(f(z)),$$ where $f(z)$ is a polynomial of degree $m^{\star}$ and prove a more general result of the above inequality for $k>1$.

We also prove that

\begin{align*}

\sup_{z \in T} \left[ \left|\frac{ {r^{*}}'(f(z))}{B^{\prime}(z)} \right| +\bigg| \frac{r'(f(z))}{B^{\prime}(z)} \bigg| \right] = \sup_{z\in T} \bigg| \frac{(rof)(z)}{f'(z)} \bigg|,

\end{align*}

and as a consequence of this result, we present a generalization of a theorem of O'Hara and Rodriguez for self-inverse 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.