Let $\mathcal{R}_n$ be the set of all rational functions of the type $r(z) = f(z)/w(z)$, where $f(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we extend some famous results concerning to the growth of polynomials by T. J. Rivlin, A. Aziz and others to the rational functions with prescribed poles and thereby obtain the analogous results for such rational functions with restricted zeros.