This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$. 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 ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ 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

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,\,p,\,n)$ were constructed recently in the work of Ariki and Koike [AK], Broué and Malle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,\,p,\,n)$ given by Ariki and Koike [AK] and Ariki [Ari].