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].

We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where $z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which $z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${\tilde A}_n$.

Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized holomorphic mappings from $B$ into $\bb{C}^n$ of ``positive real part'', is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of $B$ which arises in the study of the Loewner equation, namely the set $S^0(B)$ of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr's work, we consider subsets of $S^0(B)$ obtained by placing restrictions on the mapping from the Carath\'eodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of $S^0(B)$ as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in $S(B)$ which can be imbedded in Loewner chains, but which do not have parametric representation.

Let $Y$ be an infinite covering space of a projective manifold $M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that $f^2 e^{-\phi}$ is integrable on $X$, and define $\O_\phi(Y)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction $\O_\phi(Y) \to \O_\phi(X)$ is an isomorphism for $d$ large enough. This yields new examples of Riemann surfaces and domains of holomorphy in $\C^n$ with corona. We consider the important special case when $Y$ is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\B$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\B$.

We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\leq m$, $n$, respectively, then there exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq \frac{1}{4}\exp (-\frac{13}{\log \lambda })$ such that for $r\in \mathcal{S}$, \[ \max_{|z| =r}| R(z)| / \min_{|z| =r} | R(z) |\leq \lambda ^{m+n}. \] Here, one may not replace $\frac{1}{4}\exp ( -\frac{13}{\log \lambda })$ by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Pad\'{e} approximants for functions meromorphic in the unit ball.

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Kor\'anyi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in $\C^2$ is studied in details.

Following the investigations of B.~Abrahamse [1], F.~Forelli [11], M.~Heins [14] and others, we continue the study of the Pick-Nevanlinna interpolation problem in multiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.

In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta$, onto a simply connected domain $\Omega$ containing infinity and which are solutions of the system of elliptic partial differential equations $\fzbb = a(z)f_z(z)$ where the second dilatation function $a(z)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

Let $m$ be a point of the maximal ideal space of $\papa$ with nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra of $\papa$. We characterize the points $m$ for which $L_m$ is a homeomorphism in terms of interpolating sequences, and we show that in this case $\papa \circ L_m$ coincides with $\papa$. Also, if $I_m$ is the ideal of functions in $\papa$ that identically vanish on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.

We construct meromorphic functions with asymptotic power series expansion in $z^{-1}$ at $\infty$ on an Arakelyan set $A$ having prescribed zeros and poles outside $A$. We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

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

We study $\hbox{\Bbbvii Z}_p$ actions on compact connected Riemann surfaces via their associated Eichler traces. We determine the set of possible Eichler traces and determine the relationship between 2 actions if they have the same trace.

This paper studies conditions on an analytic function that imply it belongs to ${\cal M}_\alpha$, the set of multipliers of the family of functions given by $f(z) = \int_{|\zeta|=1} {1 \over (1-\overline\zeta z)^\alpha} \,d\mu (\zeta)$ $(|z|<1)$ where $\mu$ is a complex Borel measure on the unit circle and $\alpha >0$. There are two main theorems. The first asserts that if $0<\alpha<1$ and $\sup_{|\zeta|=1} \int^1_0 |f'(r\zeta)| (1-r)^{\alpha-1} \,dr<\infty$ then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha \leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {|f(e^{i(t+s)}) - 2f(e^{it}) + f(e^{i(t-s)})| \over s^{2-\alpha}} \, ds < \infty$ then $f \in {\cal M}_\alpha$. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

Cauchy and Poisson integrals over {\it unbounded\/} sets are employed to prove Mittag-Leffler type theorems with massive singularities as well as approximation theorems for holomorphic and harmonic functions.

For a fixed algebraic number $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that \[ \min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha -\beta| \asymp \frac{1}{\alpha^{N}}, \] and for a root of unity $\alpha$ that \[ \min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} |\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi (d)\right\rceil +1}}. \] We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly.

In the genus polynomial of the graph $G$, the coefficient of $x^k$ is the number of distinct embeddings of the graph $G$ on the oriented surface of genus $k$. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore also unimodal. The geometric distribution of the zeros of some of these polynomials is also investigated and some new genus polynomials are presented.

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous $q$-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

This paper studies closed subspaces $L$ of the Hardy spaces $H^p$ which are $g$-invariant ({\it i.e.}, $g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If $p=2$, the Wold decomposition theorem implies that there is a countable ``$g$-basis'' $f_1, f_2,\ldots$ of $L$ in the sense that $L$ is a direct sum of spaces $f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$. The basis elements $f_j$ satisfy the additional property that $\int_T |f_j|^2 g^k=0$, $k=1,2,\ldots\,.$ We call such functions $g$-$2$-inner. It also follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot (F_2\circ g)$ where $h_{f,2}$ is $g$-$2$-inner and $F$ is outer, generalizing the classical Riesz factorization. Using $L^p$ estimates for the canonical decomposition of $H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite Blaschke product we obtain, for any $g$-invariant $L\subseteq H^p$, a finite $g$-basis of $g$-$p$-inner functions.

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 $p=2$.