Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation~(P$_6$) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and B\"acklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlev\'e equations (P$_1$)--(P$_5$) below, which have been investigated by many authors [1]--[7].

This paper considers the rational system ${\cal P}_n (a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)}, P\in {\cal P}_n\bigr\}$ with nonreal elements in $\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [-1,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results.