Computational Methods and Function Theory
Proceedings 1994

Abstracts

B. Aebischer: Stable Convergence of Sequences of Möbius Transformations, 001--022
The search for conditions which guarantee convergence of a parameter depending continued fraction in a neighborhood of a parameter value leads to a natural notion of stable convergence. Assume the sequence of linear fractional transformations $S_n=s_1\circ\dots\circ s_n$, $s_n(x) = \frac{a_n x+b_n}{c_n x+d_n}$, converges at some point $x_0\in \hat{\Bbb C}$. (For continued fractions $x_0=0$, $s_n(x) = \frac{a_n}{b_n+x}$.) Roughly, the convergence is called stable if for any choice of linear fractional transformations $\tilde s_n$ sufficiently close to $s_n$, the sequence $\tilde S_n(x_0)$ converges, where $\tilde S_n = \tilde s_1\circ\dots\circ \tilde s_n$. For instance, a C-fraction ${\bf K}(\frac{a_n z^{\alpha_n}}{1})$ ($a_n \not=0$) which converges stably at $z_0 \in {\Bbb C}\setminus\{0\}$ will converge in a neighborhood of $z_0$, provided $\alpha_n$ is bounded. Similar assertions hold for general T-fractions, associated continued fractions, and J-fractions.
If a subset of the M\"obius group satisfies a certain contraction condition, then every sequence generated by elements of the set converges stably. This result has many consequences, especially for limit periodic continued fractions. Other applications include stability versions of the theorem of Prings\-heim, the theorem of Worpitzky, and the uniform parabola theorem. A weaker version of stability is shown to imply stable convergence provided the $s_n$ are contained in a compact subset of the M\"obius group.
The results given (as well as the methods used) also hold for M\"obius transformations in higher dimension.

R. M. Ali, V. Singh: Coefficients of Parabolic Starlike Functions of Order r, 023--036
For $0\leq\rho < 1$ let $\Omega _\rho$ be the parabolic region $\Omega _\rho =\{u+iv\,:\,v^2\leq 4 (1-\rho )(u-\rho )\}=\{w\,:\,|w-1|\leq1-2\rho +\RE w\}$. The class of normalized analytic functions $f$ in the unit disk $U$ for which $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $zf'(z)/f(z)\in\Omega _\rho ,\ z\in U$, is denoted by $S_p(\rho )$ and is called the class of parabolic starlike functions of order $\rho$. For $\rho =\frac{1}{2}$ this class of functions arises in the study of uniformly covex functions introduced by Goodman and studied in detail by W. Ma \& D. Minda, and F. R\o nning.
In the present paper we obtain sharp upper bounds on the coefficients $|a_n|$, $n=2,3$ and $4$ and the first four coefficients of functions inverse to $f\in\Sp$. Unlike in the case of $\rho =\frac{1}{2}$, in the present situation there are at least two extremal functions, and for the fourth coefficient of the inverse functions, three different extremal functions arise. If $\frac{zf'(z)}{f(z)}=1+\sum_{k=1}^{\infty}b_kz^k$ and $f\in\Sp$, we also obtain a sharp estimate on $\sum_{k=1}^{\infty}|b_k|^2$, which yields a general Littlewood type bound on $|a_n|$.

M. K. Aouf, M. P. Chen: On Certain Classes of Multivalent Functions Defined by the Ruscheweyh Derivative, 037--048
Let $M_{n, p}(A, B, \alpha)$ denote the class of functions $f$ of the form $f(z) = z^p + \sum_{k=1}^\infty a_{p+k}z^{p+k}$ which are regular in the unit disc $\D = \{z:\,|z| < 1\}$ and satisfy the condition
$$\frac{{\cal D}^{n+p}f(z)}{z^p} \prec \frac{1+[B+(A-B)(1-\frac{\alpha}{p})]z}{1+Bz}, \quad z \in \D,$$
where $-1 \leq B < A \leq 1$, $0 \leq \alpha < p$ and $n$ is an integer such that $n > -p$ and
$${\cal D}^{n+p-1}f(z) = \frac{z^p(z^{n-1}f(z))^{(n+p-1)}}{(n+p-1)!}.$$
In this paper we show that the functions in $M_{n,p}(A, B, \alpha)$ are $p$-valent. Then we obtain sharp coefficient estimates, maximization of $| a_{p+2} - \mu a^2_{p+1}|$ and a closure theorem for the class $M_{n,p}(A,B,\alpha)$. We also obtain a sufficient condition, in terms of coefficients, for a function to be in $M_{n,p}(A,B,\alpha)$ when $-1 \le B < 0$.

I. N. Baker, R. Maalouf: Convergence of a Modified Iteration Process, 049--056
A general result on families of analytic functions with a common invariant domain is applied to extend previous results on the convergence of infinite exponentials with arbitrary complex exponents.

A. F. Beardon: Iteration of Analytic Euclidean Contractions, 057--074
Let $D$ be a bounded convex domain and suppose that $f$ is an analytic map of $D$ into itself with $|f'| < 1$. Then the diameter of $f^n(D)$ converges to zero and we obtain upper and lower estimates on the rate of this convergence. The methods used also give results in other circumstances; for example, for contractions of a domain in $n$-dimensional Euclidean space.

H.-P. Blatt: The Distribution of Sign Changes of the Error Function in Best Lp-approximation, 075--088
For a continuous $2\pi$-periodic real-valued function $f$, we investigate the asymptotic behavior of the zeros of the error function of best $L^2$-approximation by trigonometric polynomials. The method of proof can also be applied for $L^p$-approximation, $p>1$, by algebraic polynomials.

M. Bonk, D. Minda, H. Yanagihara: The Hyperbolic Metric on Bloch Regions, 089--100
A region $\Omega$ in the complex plane is called a Bloch region provided it does not contain arbitrarily large disks. We investigate the behavior of the density $\lambda_\Omega(w)$ of the hyperbolic metric on Bloch regions. A number of sharp inequalities involving $\lambda_\Omega(w)$ and its gradient are obtained from results for locally univalent Bloch functions. The extremal region for all of the inequalities is a punctured disk. In addition, we show that $1/\lambda_\Omega(w)$ satisfies a H\"older condition of order $\beta$ for some (all) $\beta \in (0,1)$ if and only if $\Omega$ is a Bloch region.

G. Brown: Positivity of Some Classical Sums, 101--110
The classical trigonometric inequalities of Young and Fej\'er-Jackson are reviewed and extended in the context of current work on sums of Gegenbauer polynomials.

A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njastad: Convergence of Orthogonal Rational Functions, 111--124
Let $\{\alpha_n\}$ be a sequence of points in the open unit disk such that $\alpha_n\rightarrow\alpha$, $|\alpha| < 1$, and let $\mu$ be a positive measure on the unit circle. Let $\{\varphi_n\}$ be orthogonal rational functions obtained by orthogonalizing the base $\{B_n\}$, where
$(\frac{\overline{\alpha_m}}{|\alpha_m|}=-1$ if $\alpha_m=0$). The functions $\varphi_n(z)$ and their superstar transforms $\varphi_n^*(z) =B_n(z)\overline{\varphi_n(1/\overline{z})}$ satisfy a recurrence relation of the form
\begin{eqnarray*} \varphi_n(z) =\frac{\kappa_n}{\kappa_{n-1}} \left[\varepsilon_n\frac{z-\alpha_{n-1}} {1-\overline{\alpha_n}z}\varphi_{n-1}(z)+ \delta_n\frac{1-\overline{\alpha_{n-1}}z} {1-\overline{\alpha_n}z}\varphi_{n-1}^*(z) \right], \end{eqnarray*}
for certain constants $\kappa_n$, $\delta_n$, $\varepsilon_n$. It is shown that if
$$\sum_{m=1}^\infty \left[\left|1+\frac{\overline{\alpha_m}}{|\alpha_m|} \overline{\varepsilon_m}\right|+|\delta_m| \right] < \infty,$$
then $\{\varphi_n^*(z)\}$ converges uniformly on the closed unit disk to the function
\begin{eqnarray*} \pi_a(z)= \frac{1}{\sqrt{2\pi}}e^{-i\lambda} \frac{\sqrt{1-|\alpha|^2}}{(1-\overline{\alpha}z)}\, \exp\!\left[-\frac{1}{4\pi}\,\int_{-\pi}^\pi \frac{e^{i\theta}+z}{e^{i\theta}-z}\ln \mu'(\theta)\,d\theta \right], \quad \lambda\in{\Bbb R}. \end{eqnarray*}
The condition above reduces to the classical condition $\sum_{m=1}^\infty|\delta_m| < \infty$ when $\alpha_n=0$ for all $n$.

Y. M. Chiang: On the Complex Oscillation of   y'' + (ez - K) y = 0 and a Result of Bank, Laine and Langley, 125--134
Bank, Laine and Langley showed that when a solution of $y''+(e^z-K)y=0$ has a finite exponent of convergence on its zero-sequence, then $K=(2n+1)^2/16$, where $n$ is a non-negative integer. We give a refinement of their result by using a different approach, which allows us to consider another application.

K. Driver, H. Stahl: Simultaneous Rational Approximants to Nikishin Systems of Two Functions, 135--146
Simultaneous rational approximants (SRA) to a pair of Markov functions $(f_1,f_2)$ that forms a Nikishin system are investigated. It is well known that all multi-indices in the table of SRAs are normal and that all common denominators $Q_0$ satisfy a multiple orthogonality relation. We show that all the $Q_0$'s also satisfy an ordinary orthogonality relation, but now with respect to a varying measure. Error formulae are presented and an interlacing property proved for the finite zeros of the two error functions in the diagonal and quasi-diagonal cases.

C. H. FitzGerald, S. Gong: The Locally Biholomorphic Bloch and Marden Constants in Several Complex Variables, 147--158
This work continues a program to generalize one variable geometric function theory to several complex variables. The first mappings studied are required to be locally biholomorphic on spaces of matrices into ${\Bbb C}^n$. It is required that these mappings have a limited rate of growth as the boundary of the domain is approached. (Otherwise the Bloch constant may not exist.) These requirements specify classes of mappings. For each class of mappings there is a positive number $r$ such that, for each mapping $f$ in the class, there is a subdomain which $f$ carries onto a ball of radius $r$ in a one-to-one fashion. The supremum of such numbers $r$ is the Bloch constant for the class of mappings. For a wide range of classes, a positive lower bound is found for the Bloch constant.\\ For the last portion of the paper, the mappings are required to be holomorphic, not necessarily locally biholomorphic. For each class of mappings, there is a positive number $\rho$ such that, for each mapping $f$, there is a hyperbolic ball of radius $\rho$ in the domain of $f$ on which $f$ is one-to-one. The supremum of such numbers $\rho$ is the Marden constant for the class of mappings. From the earlier results of the paper, a positive lower bound of the Marden constant is obtained.

D. Gaier: Conformal Modules and their Computation, 159--172
We give a survey of old and new methods to determine the modulus $m(Q)$ of a quadrilateral $Q$ and the modulus $M(G)$ of a ring shaped domain $G$. We concentrate on direct methods, \ie on methods which avoid the computation of the corresponding conformal map.

F. W. Gehring, G. J. Martin: Holomorphic Motions, Schottky's Theorem and an Inequality for Discrete Groups, 173--182
We show how the theory of holomorphic motions and estimates for Schottky's Theorem can be used to extend a known inequality concerning two generator discrete groups of M\"{o}bius transformations.

V. Ya. Gutlyanskii: On the Mean Area Growth of the Schwarzian and Logarithmic Derivative in the Disk, 183--196
Let $f$ be analytic and univalent in the unit disk $\scriptstyle \Bbb D$ and maps it onto a Jordan domain in the complex plane $\scriptstyle \Bbb C$. First we show that $\int\!\!\!\int |\{f,z\}|dm_z < \infty$ implies that $g=\ln f'$ is continuous in $\overline{\scriptstyle \Bbb D}$. Then we give conditions for $f'(z)$ or $1/f'(z)$ to belong to the Hardy space $H^1$. Finally we note that if $f$ be analytic in $\scriptstyle \Bbb D$ then $\int\!\!\!\int |\{f,z\}|^{1/2}dm_z < \infty$ implies that $f$ is of bounded characteristic.

S. Koumandos: Some Positive Cotes Numbers for the Jacobi Weight Function, 197--206
Some new inequalities for sums of Legendre polynomials are used to prove the positivity of the Cotes numbers for the quadrature method of interpolating at the zeros of Jacobi polynomials and then integrating with respect to $w(x) \mbox{d}x$ on $[-1,1]$, where
$$w(x) = (1-x)^c (1+x)^d,\ c > -1,\ d > -1 \quad (c \neq 0, d \neq 0).$$
The positivity of some related sums of Jacobi polynomials is also established.

W. Lauf: Examples of Non-locally Compact Spaces S(G), 207--218
Let $G \subsetneq {\Bbb C }$ be a simply-connected domain and let $\sg$ be its group of conformal automorphisms with the topology of {\it uniform} chordal convergence on $G$.\\ In Gaier, D., \"Uber R\"aume konformer Selbstabbildungen ebener Gebiete, {\sl Math.\ Z.\/} {\bf 187} (1984), 227--257, Gaier raised the question whether each space $\sg$ is locally compact. In this paper we give examples of simply-connected domains with exactly one or two non-degenerate prime ends and {\it non}-locally compact automorphism spaces.

X. Li: On the Convergence of Double Least-Squares Inverses, 219--226
As simple applications of Szeg\H{o}'s theory of orthogonal polynomials, we generalize some C.~K.~Chui's convergence results on double least-squares inverse method that has been used to approximate a polynomial by another polynomial having no zeros in the closed unit disk.

L. Lorentzen: Pertubations of Linear Recurrence Relations, 227--242
In this paper we compare solutions $\{\tilde Z_n\}$ of a recurrence relation
$$\tilde Z_n+\tilde a_n^{(1)}\tilde Z_{n-1}+\tilde a_n^{(2)}\tilde Z_{n-2}=0; \quad\tilde a_n^{(2)}\ne 0;\qquad n=1,2,3,\dots$$
to solutions $\{Z_n\}$ of the perturbed equation
$$Z_n+a_n^{(1)}Z_{n-1}+a_n^{(2)}Z_{n-2}=0;\quad a_n^{(2)}\ne 0; \qquad n=1,2,3,\dots,$$
where $\lim(a_n^{(1)}-\tilde a_n^{(1)})=\lim(a_n^{(2)}-\tilde a_n^{(2)})=0$. In particular we find sufficient conditions for
$$\lim_{n\to\infty}(Z_n/Z_{n-1}-\tilde Z_n/\tilde Z_{n-1})=0$$
and
$$\lim_{n\to\infty}Z_n/\tilde Z_n=\alpha\qquad\mbox{for some}\,\,\,\alpha\in\Bbb C.$$
The results generalize classical results by Perron and Evgrafov.

J. Modersitzki, G. Opfer: Faber Versus Minimal Polynomials on Annular Sectors, 243--266
We are investigating the size of minimal polynomials and Faber polynomials on annular sectors. The size is the uniform norm of the corresponding polynomials on the underlying sector. If the size is less than one, the corresponding polynomial is suitable for the use in so-called polynomial-based iteration-schemes for solving linear systems. For intrinsic reasons all polynomials $p$ to be considered must satisfy the normalization condition $p(0)=1$. The annular sectors play the r\^ole of an inclusion set for the eigenvalues of the underlying matrix in the linear system to be solved. The minimal polynomials are those with least uniform norm in this class. The computation of minimal polynomials requires a Remez-type algorithm for complex cases. Faber polynomials are computed by means of Schwarz-Christoffel-formulae. Here singular integrals of different types occur. Various Gauss-formulae with high knot numbers are employed. The annular sectors have the remarkable property, that they are invariant under monomial mappings, which also allows the use of several unconnected sectors of the same size distributed equiangularly over the annulus. This is of great help in certain applications, where the mentioned eigenvalues are located in a disconnected set. There are comparisons in graphical form of both types of polynomials with respect to degree and to the different types of sectors. Most of the numerical work is included in these figures.

N. Papamichael, N. S. Stylianopoulos: Domain Decomposition for Conformal Maps, 267--292
This paper is concerned with certain aspects of the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals. Our main purpose is to make use of the theory of the method in order to investigate the quality of certain heuristic rules that we have come across in the VLSI literature, in connection with the measurement of resistance values of integrated circuit networks.

E. A. Rakhmanov, E. B. Saff, Y. M. Zhou: Electrons on the Sphere, 293--310
We investigate the energy of arrangements of $N$ points (charged particles) on the surface of the unit sphere in ${\Bbb R}^3$, interacting through a power law potential $V(r) = r^\alpha, \;\; - 2 < \alpha < 2, \;\; \alpha \neq 0$ and $V (r) = \log (1/r)$ for $\alpha = 0$. Results of numerical experiments are presented for the three classical cases $\alpha = 0 , \pm 1, \mbox{ for } N=2,\ldots,200$. We also investigate the geometric structures of the equilibrium points and their group properties. In the case $\alpha = 0$, we prove that the $N$ points of minimal energy must be well-separated in the sense that any two distinct points are at least $(3/5) (1/\sqrt{N})$ units distance apart.

G. Schmeisser: Integral Inequalities for Entire Functions of Exponential Type, 311--318
We consider a derivative estimate and a growth estimate for entire functions of exponential type in the $L^p$ norm and various modifications and refinements. Then we describe generalizations which include extensions to $p\in (0,1),$ the range where $\|.\|_p$ is no longer a norm. Next we show that all of these results can be deduced from one inequality, which is a growth estimate with a free parameter. Finally we pose a problem.

T. N. Shanmugam, V. Ravichandran: Certain Properties of Uniformly Convex Functions, 319--324
We investigate the classes of normalized functions $f(z)$ analytic in the open unit disk ${\Bbb D}=\{z:|z| < 1\}$ for which $\frac{zf'(z)}{f(z)}$ or $\frac{zf''(z)}{f'(z)}+1$ ranges over the parabolic region $\Omega=\{w:|w-1| < \Real w\}$. We also discuss some integral operators and radius problems.

H. Stahl: Simultaneous Rational Approximants, 325--350
Results about simultaneous rational approximants will be surveyed. Special attention will be paid to the approximation of Angelesco and Nikishin systems. Simultaneous rational approximants are a natural generalization of Pad\'e approximants and continued fractions. Instead of only one function now a vector $(f_1,\dots,f_m)$ of $m$ functions is simultaneously approximated by a vector $(q_1/q_0,\dots,q_m/q_0)$ of rational functions with a common denominator $q_0$. It is well known that the theory of Pad\'e approximants is closely related with orthogonal polynomials, three term recurrence relations, and Gau\ss{} quadrature. In an analoguous way the theory of simultaneous rational approximants is connected with multiple orthogonal polynomials, $m+2$-term recurrence relations, and simultaneous quadrature. After surveying the basic concepts underlying the definition of simultaneous rational approximants, we shortly review multiple orthogonal analogues to classical orthogonal polynomials. We then turn to the approximation of systems (vectors) of Markov functions. Contrary to the case of a single Markov function, where Markov's Theorem gives a very complete answer to the convergence problem, in the vector case general results are comparatively rare; such results have been proved so far only for special systems, namely the Angelesco and the Nikishin systems. Results about Angelesco systems will be reviewed and new results about Nikishin systems will be presented. In the proofs of convergence for both types of systems an equilibrium problem for vector potentials plays a key role. The connection between the vector equilibrium problem and certain algebraic functions will be discussed.