On the Cauchy index of a real rational function and the index theory of pseudo-lossless rational functions

1996 ◽  
pp. 23-31
Author(s):  
Yves V. Genin
Keyword(s):  
Rational Function ◽  
Index Theory ◽  
Rational Functions ◽  
Cauchy Index

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

The depth of continued fraction expansion for some classes of rational functions

2020 ◽  
pp. 2150033
Author(s):  
Yanapat Tongron ◽  
Narakorn Rompurk Kanasri ◽  
Vichian Laohakosol
Keyword(s):  
Rational Function ◽  
Upper Bound ◽  
Continued Fraction ◽  
Rational Functions ◽  
Linear Fractional Transformation ◽  
Continued Fraction Expansion ◽  
Polynomial Formula ◽  
Nonzero Polynomial

For nonzero polynomials [Formula: see text] and [Formula: see text] over a field [Formula: see text], let [Formula: see text] be the depth (length) of the continued fraction expansion for [Formula: see text]. An upper bound on [Formula: see text], for nonzero polynomial [Formula: see text] and rational function [Formula: see text] is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

scholarly journals Any counterexample to Makienko’s conjecture is an indecomposable continuum

2009 ◽  
Vol 29 (3) ◽  
pp. 875-883 ◽  
Author(s):  
CLINTON P. CURRY ◽  
JOHN C. MAYER ◽  
JONATHAN MEDDAUGH ◽  
JAMES T. ROGERS Jr
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Fatou Set ◽  
Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

scholarly journals Minimal models for rational functions in a dynamical setting

2012 ◽  
Vol 15 ◽  
pp. 400-417
Author(s):  
Nils Bruin ◽  
Alexander Molnar
Keyword(s):  
Rational Function ◽  
Rational Functions ◽  
Minimal Models ◽  
Linear Transformations ◽  
Single Orbit ◽  
Practical Algorithm

AbstractWe present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with eight integers in an orbit and several minimal cubic rational functions with ten integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro’s conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.

scholarly journals Simultaneous Iteration by Entire or Rational Functions and Their Inverses

1983 ◽  
Vol 34 (3) ◽  
pp. 364-367 ◽  
Author(s):  
I. N. Baker ◽  
Zalman Rubinstein
Keyword(s):  
Rational Function ◽  
Rational Functions ◽  
Simultaneous Iteration

AbstractFor a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.

Some Properties of Rational Functions with Prescribed Poles

1999 ◽  
Vol 42 (4) ◽  
pp. 417-426 ◽  
Author(s):  
Abdul Aziz-Ul-Auzeem ◽  
B. A. Zarger
Keyword(s):  
Unit Circle ◽  
Rational Function ◽  
Rational Functions ◽  
Sharp Result

AbstractLet P(z) be a polynomial of degree not exceeding n and let where |aj| > 1, j = 1, 2,…,n. If the rational function r(z) = P(z)/W(z) does not vanish in |z| < k, then for k = 1 it is known thatwhere B(Z) = W*(z)/W(z) and . In the paper we consider the case when k > 1 and obtain a sharp result. We also show thatwhere , and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

On Rational Functions Related to Algorithms for a Computation of Roots. II

2019 ◽  
Vol 7 (4) ◽  
pp. 26-29
Author(s):  
Mirosław Baran
Keyword(s):  
Rational Function ◽  
Iterative Algorithm ◽  
Direct Proof ◽  
Rational Functions ◽  
Basic Tool ◽  
Square Roots ◽  
Composition Properties

We discuss a nice composition properties related to algorithms for computation of N-roots. Our approach gives direct proof that a version of Newton's iterative algorithm is of order 2. A basic tool used in this note are properties of rational function Φ(w; z) = z-w/(z+w), which was used earlier in [1] in the case of algorithms for computations of square roots.

On Rational Functions Related to Algorithms for a Computation of Roots. I

2019 ◽  
Vol 7 (4) ◽  
pp. 17-25
Author(s):  
Mirosław Baran
Keyword(s):  
Rational Function ◽  
Iterative Algorithm ◽  
Chebyshev Polynomials ◽  
Direct Proof ◽  
Rational Functions ◽  
General Construction ◽  
Square Root ◽  
Essential Completion ◽  
Special Case ◽  
Surprising Fact

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k-th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

scholarly journals SOLUTION OF THE HURWITZ PROBLEM FOR LAURENT POLYNOMIALS

2009 ◽  
Vol 18 (02) ◽  
pp. 271-302 ◽  
Author(s):  
F. PAKOVICH
Keyword(s):  
Rational Function ◽  
Rational Functions ◽  
Single Element ◽  
Existence Problem ◽  
Laurent Polynomials ◽  
Hurwitz Problem ◽  
The One

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements.

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