It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szeg\"o polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients {\it etc.} are derived.

We study the function spaces $Q_p(R)$ defined on a Riemann surface $R$, which were earlier introduced in the unit disk of the complex plane. The nesting property $Q_p(R)\subseteq Q_q(R)$ for $0
Categories:30D45, 30D50, 30F35

For a meromorphic (or harmonic) function $f$, let us call the dilation of $f$ at $z$ the ratio of the (spherical) metric at $f(z)$ and the (hyperbolic) metric at $z$. Inequalities are known which estimate the $\sup$ norm of the dilation in terms of its $L^p$ norm, for $p>2$, while capitalizing on the symmetries of $f$. In the present paper we weaken the hypothesis by showing that such estimates persist even if the $L^p$ norms are taken only over the set of $z$ on which $f$ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that $p=2$.